Martin Escardo 2012. 

\begin{code}

{-# OPTIONS --without-K #-}

module FirstProjectionInjective where

open import Equality
open import SetsAndFunctions

π₀-mono : {X : Set} → {Y : X → Set} → (∀{x : X} → ∀(y y' : Y x) → y ≡ y')

   → ∀(u v : Σ \(x : X) → Y x) → π₀ u ≡ π₀ v → u ≡ v

π₀-mono {X} {Y} f u v r =
 lemma r (π₁ u) (π₁ v) (f (subst {X} {Y} r (π₁ u)) (π₁ v))
 where
  A : ∀(x x' : X) → x ≡ x' → Set
  A x x' r = ∀(y : Y x) → ∀(y' : Y x') → subst r y ≡ y' → (x , y) ≡ (x' , y')

  lemma : ∀{x x' : X} → ∀(r : x ≡ x') → A x x' r
  lemma = J A (λ x x' y → cong (λ y → (x , y)))

\end{code}

I learned this useful theorem from Thierry Coquand. The idea is
that if the premise holds, this means that Σ \(x : X) → Y x) can
be considered as the subobject of X of elements satisfying Y x,
with the first projection as the monomorphic inclusion into X.