Theorem swapf2fvala 50717

Description: The morphism part of the swap functor. See also swapf2fval 50718. (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 50715	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
7	6	fveq2d 6920	. 2 ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (2nd ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩))
8	4	fvexi 6930	. . . 4 ⊢ 𝐵 ∈ V
9	8	mptex 7260	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}) ∈ V
10	8, 8	mpoex 8127	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})) ∈ V
11	9, 10	op2nd 8044	. 2 ⊢ (2nd ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))
12	7, 11	eqtrdi 2814	1 ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {csn 4588  ⟨cop 4840  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5657  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  2^nd c2nd 8034  Basecbs 17431  Hom chom 17483   ×_c cxpc 18392   swap_F cswapf 50712

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-swapf 50713

This theorem is used by:  swapf2fval  50718