Theorem swapf2fvala 50075

Description: The morphism part of the swap functor. See also swapf2fval 50076. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapf2fvala.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf2fvala.b	⊢ 𝐵 = (Base‘𝑆)
swapf2fvala.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))

Assertion

Ref	Expression
swapf2fvala	⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))

Distinct variable groups:   𝑢,𝐵,𝑣   𝑢,𝐶,𝑣   𝑢,𝐷,𝑣   𝑓,𝐻,𝑢,𝑣   𝑢,𝑆,𝑣   𝜑,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑆(𝑓)   𝑈(𝑣, 𝑢, 𝑓)   𝑉(𝑣, 𝑢, 𝑓)

Proof of Theorem swapf2fvala

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swapfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
2		swapfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
3		swapf2fvala.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
4		swapf2fvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
5		swapf2fvala.h	. . . 4 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
6	1, 2, 3, 4, 5	swapfval 50073	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
7	6	fveq2d 6889	. 2 ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (2nd ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩))
8	4	fvexi 6899	. . . 4 ⊢ 𝐵 ∈ V
9	8	mptex 7225	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}) ∈ V
10	8, 8	mpoex 8078	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})) ∈ V
11	9, 10	op2nd 7997	. 2 ⊢ (2nd ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))
12	7, 11	eqtrdi 2816	1 ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {csn 4591  ⟨cop 4597  ∪ cuni 4874   ↦ cmpt 5194  ^◡ccnv 5662  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  2^nd c2nd 7987  Basecbs 17287  Hom chom 17339   ×_c cxpc 18242   swap_F cswapf 50070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-swapf 50071

This theorem is used by:  swapf2fval  50076