Theorem swapf2fvala 50493

Description: The morphism part of the swap functor. See also swapf2fval 50494. (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 50491	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩)
7	6	fveq2d 6892	. 2 ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (2nd ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩))
8	4	fvexi 6902	. . . 4 ⊢ 𝐵 ∈ V
9	8	mptex 7231	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}) ∈ V
10	8, 8	mpoex 8094	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})) ∈ V
11	9, 10	op2nd 8012	. 2 ⊢ (2nd ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))⟩) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}))
12	7, 11	eqtrdi 2811	1 ⊢ (𝜑 → (2nd ‘(𝐶 swapF 𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {csn 4584  ⟨cop 4834  ∪ cuni 4866   ↦ cmpt 5185  ^◡ccnv 5649  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424  2^nd c2nd 8002  Basecbs 17394  Hom chom 17446   ×_c cxpc 18349   swap_F cswapf 50488

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-swapf 50489

This theorem is used by:  swapf2fval  50494