Theorem swapf2vala 49928

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

Hypotheses

Ref	Expression
swapf1a.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨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 49906	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5	1, 2, 3	elxpcbasex2 49908	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
6		swapf2a.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
7		swapf1a.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
8	4, 5, 1, 2, 6, 7	swapf2fval 49923	. 2 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
9		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑢 = 𝑋)
10		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑣 = 𝑌)
11	9, 10	oveq12d 7426	. . 3 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑢𝐻𝑣) = (𝑋𝐻𝑌))
12	11	mpteq1d 5202	. 2 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
13		swapf2a.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7441	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
15	14	mptex 7219	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V)
17	8, 12, 3, 13, 16	ovmpod 7560	1 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {csn 4591  ⟨cop 4597  ∪ cuni 4873   ↦ cmpt 5193  ^◡ccnv 5658  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  Hom chom 17317   ×_c cxpc 18220   swap_F cswapf 49917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-1cn 11154  ax-addcl 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-nn 12230  df-slot 17238  df-ndx 17250  df-base 17266  df-xpc 18224  df-swapf 49918

This theorem is referenced by:  swapf2a  49929  swapf2val  49931  swapf2f1oaALT  49936