Metamath Proof Explorer

Theorem cocanfo

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011) (Proof shortened by Mario Carneiro, 27-Dec-2014)

Ref Expression
Assertion cocanfo ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → 𝐺 = 𝐻 )

Proof

Step Hyp Ref Expression
1 simplr ⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) )
2 1 fveq1d ⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑦 ) )
3 simpl1 ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → 𝐹 : 𝐴 –onto→ 𝐵 )
4 fof ⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
5 3 4 syl ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 )
6 fvco3 ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) )
7 5 6 sylan ⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) )
8 fvco3 ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) )
9 5 8 sylan ⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐻 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) )
10 2 7 9 3eqtr3d ⊢ ( ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) )
11 10 ralrimiva ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) )
12 fveq2 ⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑥 → ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐺 ‘ 𝑥 ) )
13 fveq2 ⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑥 → ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐻 ‘ 𝑥 ) )
14 12 13 eqeq12d ⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑥 → ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) )
15 14 cbvfo ⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ( ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) )
16 3 15 syl ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → ( ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) = ( 𝐻 ‘ ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) )
17 11 16 mpbid ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → ∀ 𝑥 ∈ 𝐵 ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) )
18 eqfnfv ⊢ ( ( 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) → ( 𝐺 = 𝐻 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) )
19 18 3adant1 ⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) → ( 𝐺 = 𝐻 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) )
20 19 adantr ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → ( 𝐺 = 𝐻 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝐺 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) )
21 17 20 mpbird ⊢ ( ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵 ) ∧ ( 𝐺 ∘ 𝐹 ) = ( 𝐻 ∘ 𝐹 ) ) → 𝐺 = 𝐻 )