Theorem swapf2vala 49275

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 49253	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5	1, 2, 3	elxpcbasex2 49255	. . 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 49270	. 2 ⊢ (𝜑 → 𝑃 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓})))
9		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑢 = 𝑋)
10		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → 𝑣 = 𝑌)
11	9, 10	oveq12d 7371	. . 3 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑢𝐻𝑣) = (𝑋𝐻𝑌))
12	11	mpteq1d 5185	. 2 ⊢ ((𝜑 ∧ (𝑢 = 𝑋 ∧ 𝑣 = 𝑌)) → (𝑓 ∈ (𝑢𝐻𝑣) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
13		swapf2a.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14		ovex 7386	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
15	14	mptex 7163	. . 3 ⊢ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) ∈ V)
17	8, 12, 3, 13, 16	ovmpod 7505	1 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579  ⟨cop 4585  ∪ cuni 4861   ↦ cmpt 5176  ^◡ccnv 5622  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  Hom chom 17191   ×_c cxpc 18093   swap_F cswapf 49264

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-1cn 11086  ax-addcl 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12148  df-slot 17112  df-ndx 17124  df-base 17140  df-xpc 18097  df-swapf 49265

This theorem is referenced by:  swapf2a  49276  swapf2val  49278  swapf2f1oaALT  49283