Theorem swapf2fval 48944

Description: The morphism part of the swap functor. See also swapf2fvala 48943. (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))
swapf2fval.o	⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)

Assertion

Ref	Expression
swapf2fval	⊢ (𝜑 → 𝑃 = (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)   𝑃(𝑣,𝑢,𝑓)   𝑆(𝑓)   𝑈(𝑣,𝑢,𝑓)   𝑂(𝑣,𝑢,𝑓)   𝑉(𝑣,𝑢,𝑓)

Proof of Theorem swapf2fval

Step	Hyp	Ref	Expression
1		swapf2fval.o	. . 3 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
2	1	fveq2d 6908	. 2 ⊢ (𝜑 → (2nd ‘(𝐶swapF𝐷)) = (2nd ‘⟨𝑂, 𝑃⟩))
3		swapfval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
4		swapfval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		swapf2fvala.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
6		swapf2fvala.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
7		swapf2fvala.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
8	3, 4, 5, 6, 7	swapf2fvala 48943	. 2 ⊢ (𝜑 → (2nd ‘(𝐶swapF𝐷)) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
9	3, 4	swapfelvv 48942	. . . 4 ⊢ (𝜑 → (𝐶swapF𝐷) ∈ (V × V))
10	1, 9	eqeltrrd 2841	. . 3 ⊢ (𝜑 → ⟨𝑂, 𝑃⟩ ∈ (V × V))
11		opelxp 5719	. . . 4 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) ↔ (𝑂 ∈ V ∧ 𝑃 ∈ V))
12	11	biimpi 216	. . 3 ⊢ (⟨𝑂, 𝑃⟩ ∈ (V × V) → (𝑂 ∈ V ∧ 𝑃 ∈ V))
13		op2ndg 8023	. . 3 ⊢ ((𝑂 ∈ V ∧ 𝑃 ∈ V) → (2nd ‘⟨𝑂, 𝑃⟩) = 𝑃)
14	10, 12, 13	3syl 18	. 2 ⊢ (𝜑 → (2nd ‘⟨𝑂, 𝑃⟩) = 𝑃)
15	2, 8, 14	3eqtr3rd 2785	1 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3479  {csn 4624  ⟨cop 4630  ∪ cuni 4905   ↦ cmpt 5223   × cxp 5681  ^◡ccnv 5682  ‘cfv 6559  (class class class)co 7429   ∈ cmpo 7431  2^nd c2nd 8009  Basecbs 17243  Hom chom 17304   ×_c cxpc 18209  swap_Fcswapf 48938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-swapf 48939

This theorem is referenced by:  swapf2fn  48947  swapf2vala  48949