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Metamath Proof Explorer

Theorem invinv

Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of Adamek p. 29. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses invfval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
invfval.n ⊢ 𝑁 = ( Inv ‘ 𝐶 )
invfval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
invss.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
invss.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
isoval.n ⊢ 𝐼 = ( Iso ‘ 𝐶 )
invinv.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
Assertion invinv ( 𝜑 → ( ( 𝑌 𝑁 𝑋 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 invfval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
2 invfval.n ⊢ 𝑁 = ( Inv ‘ 𝐶 )
3 invfval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
4 invss.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 invss.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 isoval.n ⊢ 𝐼 = ( Iso ‘ 𝐶 )
7 invinv.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
8 1 2 3 4 5 invsym2 ⊢ ( 𝜑 → ◡ ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝑁 𝑋 ) )
9 8 fveq1d ⊢ ( 𝜑 → ( ◡ ( 𝑋 𝑁 𝑌 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = ( ( 𝑌 𝑁 𝑋 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) )
10 1 2 3 4 5 6 invf1o ⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) )
11 f1ocnvfv1 ⊢ ( ( ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) ∧ 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) ) → ( ◡ ( 𝑋 𝑁 𝑌 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )
12 10 7 11 syl2anc ⊢ ( 𝜑 → ( ◡ ( 𝑋 𝑁 𝑌 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )
13 9 12 eqtr3d ⊢ ( 𝜑 → ( ( 𝑌 𝑁 𝑋 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )