invf

Description: The inverse relation is a function from isomorphisms to isomorphisms. (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	invf	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) ⟶ ( 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	invfun	⊢ ( 𝜑 → Fun ( 𝑋 𝑁 𝑌 ) )
8	7	funfnd	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) Fn dom ( 𝑋 𝑁 𝑌 ) )
9	1 2 3 4 5 6	isoval	⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) = dom ( 𝑋 𝑁 𝑌 ) )
10	9	fneq2d	⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) Fn ( 𝑋 𝐼 𝑌 ) ↔ ( 𝑋 𝑁 𝑌 ) Fn dom ( 𝑋 𝑁 𝑌 ) ) )
11	8 10	mpbird	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) Fn ( 𝑋 𝐼 𝑌 ) )
12		df-rn	⊢ ran ( 𝑋 𝑁 𝑌 ) = dom ^◡ ( 𝑋 𝑁 𝑌 )
13	1 2 3 4 5	invsym2	⊢ ( 𝜑 → ^◡ ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝑁 𝑋 ) )
14	13	dmeqd	⊢ ( 𝜑 → dom ^◡ ( 𝑋 𝑁 𝑌 ) = dom ( 𝑌 𝑁 𝑋 ) )
15	1 2 3 5 4 6	isoval	⊢ ( 𝜑 → ( 𝑌 𝐼 𝑋 ) = dom ( 𝑌 𝑁 𝑋 ) )
16	14 15	eqtr4d	⊢ ( 𝜑 → dom ^◡ ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝐼 𝑋 ) )
17	12 16	eqtrid	⊢ ( 𝜑 → ran ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝐼 𝑋 ) )
18		eqimss	⊢ ( ran ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝐼 𝑋 ) → ran ( 𝑋 𝑁 𝑌 ) ⊆ ( 𝑌 𝐼 𝑋 ) )
19	17 18	syl	⊢ ( 𝜑 → ran ( 𝑋 𝑁 𝑌 ) ⊆ ( 𝑌 𝐼 𝑋 ) )
20		df-f	⊢ ( ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) ⟶ ( 𝑌 𝐼 𝑋 ) ↔ ( ( 𝑋 𝑁 𝑌 ) Fn ( 𝑋 𝐼 𝑌 ) ∧ ran ( 𝑋 𝑁 𝑌 ) ⊆ ( 𝑌 𝐼 𝑋 ) ) )
21	11 19 20	sylanbrc	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) ⟶ ( 𝑌 𝐼 𝑋 ) )

Metamath Proof Explorer

Theorem invf

Proof