Theorem invf1o

Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism F e. ( X I Y ) has a unique inverse, denoted by ( ( InvC )F ) . Remark 3.12 of Adamek p. 28. (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 )
	invfval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	invfval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	isoval.n	⊢ 𝐼 = ( Iso ‘ 𝐶 )
Assertion	invf1o	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		invfval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		invfval.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
3		invfval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		invfval.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		invfval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		isoval.n	⊢ 𝐼 = ( Iso ‘ 𝐶 )
7	1 2 3 4 5 6	invf	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) ⟶ ( 𝑌 𝐼 𝑋 ) )
8	7	ffnd	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) Fn ( 𝑋 𝐼 𝑌 ) )
9	1 2 3 5 4 6	invf	⊢ ( 𝜑 → ( 𝑌 𝑁 𝑋 ) : ( 𝑌 𝐼 𝑋 ) ⟶ ( 𝑋 𝐼 𝑌 ) )
10	9	ffnd	⊢ ( 𝜑 → ( 𝑌 𝑁 𝑋 ) Fn ( 𝑌 𝐼 𝑋 ) )
11	1 2 3 4 5	invsym2	⊢ ( 𝜑 → ◡ ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝑁 𝑋 ) )
12	11	fneq1d	⊢ ( 𝜑 → ( ◡ ( 𝑋 𝑁 𝑌 ) Fn ( 𝑌 𝐼 𝑋 ) ↔ ( 𝑌 𝑁 𝑋 ) Fn ( 𝑌 𝐼 𝑋 ) ) )
13	10 12	mpbird	⊢ ( 𝜑 → ◡ ( 𝑋 𝑁 𝑌 ) Fn ( 𝑌 𝐼 𝑋 ) )
14		dff1o4	⊢ ( ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) ↔ ( ( 𝑋 𝑁 𝑌 ) Fn ( 𝑋 𝐼 𝑌 ) ∧ ◡ ( 𝑋 𝑁 𝑌 ) Fn ( 𝑌 𝐼 𝑋 ) ) )
15	8 13 14	sylanbrc	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) )