Theorem swapf2vala 49757

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 49735	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5	1, 2, 3	elxpcbasex2 49737	. . 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 49752	. 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 7378	. . 3 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑢𝐻𝑣) = (𝑋𝐻𝑌))
12	11	mpteq1d 5176	. 2 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
13		swapf2a.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7393	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
15	14	mptex 7171	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V)
17	8, 12, 3, 13, 16	ovmpod 7512	1 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {csn 4568  ⟨cop 4574  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5623  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Hom chom 17222   ×_c cxpc 18125   swap_F cswapf 49746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-nn 12166  df-slot 17143  df-ndx 17155  df-base 17171  df-xpc 18129  df-swapf 49747

This theorem is referenced by:  swapf2a  49758  swapf2val  49760  swapf2f1oaALT  49765