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Metamath Proof Explorer

Theorem f1cofveqaeqALT

Description: Alternate proof of f1cofveqaeq , 1 essential step shorter, but having more bytes (305 versus 282). (Contributed by AV, 3-Feb-2021) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion f1cofveqaeqALT ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) → 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 f1f ⊢ ( 𝐺 : 𝐴 –1-1→ 𝐵 → 𝐺 : 𝐴 ⟶ 𝐵 )
2 fvco3 ⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) )
3 2 adantrr ⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) )
4 fvco3 ⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) )
5 4 adantrl ⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) )
6 3 5 eqeq12d ⊢ ( ( 𝐺 : 𝐴 ⟶ 𝐵 ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) ) )
7 6 ex ⊢ ( 𝐺 : 𝐴 ⟶ 𝐵 → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) )
8 1 7 syl ⊢ ( 𝐺 : 𝐴 –1-1→ 𝐵 → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) )
9 8 adantl ⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) → ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) ) ) )
10 9 imp ⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) ↔ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) ) )
11 f1co ⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) → ( 𝐹 ∘ 𝐺 ) : 𝐴 –1-1→ 𝐶 )
12 f1veqaeq ⊢ ( ( ( 𝐹 ∘ 𝐺 ) : 𝐴 –1-1→ 𝐶 ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) → 𝑋 = 𝑌 ) )
13 11 12 sylan ⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑌 ) → 𝑋 = 𝑌 ) )
14 10 13 sylbird ⊢ ( ( ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐺 : 𝐴 –1-1→ 𝐵 ) ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑌 ) ) → 𝑋 = 𝑌 ) )