Theorem swapf2vala 48949

Description: The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapf1a.o	⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
swapf1a.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf1a.b	⊢ 𝐵 = (Base‘𝑆)
swapf1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapf2a.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
swapf2a.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))

Assertion

Ref	Expression
swapf2vala	⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Distinct variable groups:   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑃(𝑓)   𝑆(𝑓)   𝑂(𝑓)

Proof of Theorem swapf2vala

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		swapf1a.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
2		swapf1a.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
3		swapf1a.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4	1, 2, 3	elxpcbasex1 48927	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5	1, 2, 3	elxpcbasex2 48929	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
6		swapf2a.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
7		swapf1a.o	. . 3 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
8	4, 5, 1, 2, 6, 7	swapf2fval 48944	. 2 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
9		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑢 = 𝑋)
10		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑣 = 𝑌)
11	9, 10	oveq12d 7447	. . 3 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑢𝐻𝑣) = (𝑋𝐻𝑌))
12	11	mpteq1d 5235	. 2 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
13		swapf2a.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7462	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
15	14	mptex 7241	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V)
17	8, 12, 3, 13, 16	ovmpod 7582	1 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3479  {csn 4624  ⟨cop 4630  ∪ cuni 4905   ↦ cmpt 5223  ^◡ccnv 5682  ‘cfv 6559  (class class class)co 7429  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  ax-cnex 11207  ax-1cn 11209  ax-addcl 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  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-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  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-om 7884  df-1st 8010  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-nn 12263  df-slot 17215  df-ndx 17227  df-base 17244  df-xpc 18213  df-swapf 48939

This theorem is referenced by:  swapf2a  48950  swapf2val  48952  swapf2f1oaALT  48957