Theorem swapf2f1o

Description: The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 8-Oct-2025)

	Ref	Expression
Hypotheses	swapf1f1o.o	⊢ ( 𝜑 → ( 𝐶 swapF 𝐷 ) = ⟨ 𝑂 , 𝑃 ⟩ )
	swapf1f1o.s	⊢ 𝑆 = ( 𝐶 ×c 𝐷 )
	swapf1f1o.t	⊢ 𝑇 = ( 𝐷 ×c 𝐶 )
	swapf2f1o.h	⊢ 𝐻 = ( Hom ‘ 𝑆 )
	swapf2f1o.j	⊢ 𝐽 = ( Hom ‘ 𝑇 )
	swapf2f1o.x	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
	swapf2f1o.y	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐷 ) )
	swapf2f1o.z	⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐶 ) )
	swapf2f1o.w	⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐷 ) )
Assertion	swapf2f1o	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑃 ⟨ 𝑍 , 𝑊 ⟩ ) : ( ⟨ 𝑋 , 𝑌 ⟩ 𝐻 ⟨ 𝑍 , 𝑊 ⟩ ) –1-1-onto→ ( ⟨ 𝑌 , 𝑋 ⟩ 𝐽 ⟨ 𝑊 , 𝑍 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		swapf1f1o.o	⊢ ( 𝜑 → ( 𝐶 swapF 𝐷 ) = ⟨ 𝑂 , 𝑃 ⟩ )
2		swapf1f1o.s	⊢ 𝑆 = ( 𝐶 ×c 𝐷 )
3		swapf1f1o.t	⊢ 𝑇 = ( 𝐷 ×c 𝐶 )
4		swapf2f1o.h	⊢ 𝐻 = ( Hom ‘ 𝑆 )
5		swapf2f1o.j	⊢ 𝐽 = ( Hom ‘ 𝑇 )
6		swapf2f1o.x	⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
7		swapf2f1o.y	⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐷 ) )
8		swapf2f1o.z	⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐶 ) )
9		swapf2f1o.w	⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐷 ) )
10		eqid	⊢ ( 𝑓 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) ↦ ∪ ◡ { 𝑓 } ) = ( 𝑓 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) ↦ ∪ ◡ { 𝑓 } )
11	10	xpcomf1o	⊢ ( 𝑓 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) ↦ ∪ ◡ { 𝑓 } ) : ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) –1-1-onto→ ( ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) )
12	4	a1i	⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝑆 ) )
13	1 6 7 8 9 2 12	swapf2val	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑃 ⟨ 𝑍 , 𝑊 ⟩ ) = ( 𝑓 ∈ ( ⟨ 𝑋 , 𝑌 ⟩ 𝐻 ⟨ 𝑍 , 𝑊 ⟩ ) ↦ ∪ ◡ { 𝑓 } ) )
14		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
15		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
16		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
17		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
18	2 14 15 16 17 6 7 8 9 4	xpchom2	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝐻 ⟨ 𝑍 , 𝑊 ⟩ ) = ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) )
19	18	mpteq1d	⊢ ( 𝜑 → ( 𝑓 ∈ ( ⟨ 𝑋 , 𝑌 ⟩ 𝐻 ⟨ 𝑍 , 𝑊 ⟩ ) ↦ ∪ ◡ { 𝑓 } ) = ( 𝑓 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) ↦ ∪ ◡ { 𝑓 } ) )
20	13 19	eqtrd	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑃 ⟨ 𝑍 , 𝑊 ⟩ ) = ( 𝑓 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) ↦ ∪ ◡ { 𝑓 } ) )
21	3 15 14 17 16 7 6 9 8 5	xpchom2	⊢ ( 𝜑 → ( ⟨ 𝑌 , 𝑋 ⟩ 𝐽 ⟨ 𝑊 , 𝑍 ⟩ ) = ( ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) ) )
22	20 18 21	f1oeq123d	⊢ ( 𝜑 → ( ( ⟨ 𝑋 , 𝑌 ⟩ 𝑃 ⟨ 𝑍 , 𝑊 ⟩ ) : ( ⟨ 𝑋 , 𝑌 ⟩ 𝐻 ⟨ 𝑍 , 𝑊 ⟩ ) –1-1-onto→ ( ⟨ 𝑌 , 𝑋 ⟩ 𝐽 ⟨ 𝑊 , 𝑍 ⟩ ) ↔ ( 𝑓 ∈ ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) ↦ ∪ ◡ { 𝑓 } ) : ( ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) × ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) ) –1-1-onto→ ( ( 𝑌 ( Hom ‘ 𝐷 ) 𝑊 ) × ( 𝑋 ( Hom ‘ 𝐶 ) 𝑍 ) ) ) )
23	11 22	mpbiri	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑃 ⟨ 𝑍 , 𝑊 ⟩ ) : ( ⟨ 𝑋 , 𝑌 ⟩ 𝐻 ⟨ 𝑍 , 𝑊 ⟩ ) –1-1-onto→ ( ⟨ 𝑌 , 𝑋 ⟩ 𝐽 ⟨ 𝑊 , 𝑍 ⟩ ) )