Theorem swapf2a 49929

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))
swapf2a.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
swapf2a	⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

Proof of Theorem swapf2a

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swapf1a.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
2		swapf1a.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
3		swapf1a.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
4		swapf1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		swapf2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		swapf2a.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
7	1, 2, 3, 4, 5, 6	swapf2vala 49928	. 2 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
8		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9	8	sneqd 4603	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → {𝑓} = {𝐹})
10	9	cnveqd 5859	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡{𝑓} = ◡{𝐹})
11	10	unieqd 4886	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ∪ ◡{𝐹})
12		swapf2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
13	6	oveqd 7425	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝑆)𝑌))
14		eqid 2769	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2769	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		eqid 2769	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
17	2, 3, 14, 15, 16, 4, 5	xpchom 18232	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝑆)𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
18	13, 17	eqtrd 2804	. . . . . 6 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
19	12, 18	eleqtrd 2871	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
20		2nd1st 8031	. . . . 5 ⊢ (𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
21	19, 20	syl 18	. . . 4 ⊢ (𝜑 → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
22	21	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
23	11, 22	eqtrd 2804	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
24		opex 5443	. . 3 ⊢ ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V)
26	7, 23, 12, 25	fvmptd 6995	1 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {csn 4591  ⟨cop 4597  ∪ cuni 4873   × cxp 5657  ^◡ccnv 5658  ‘cfv 6534  (class class class)co 7408  1^st c1st 7980  2^nd c2nd 7981  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-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

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-nel 3071  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-riota 7365  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-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-hom 17330  df-cco 17331  df-xpc 18224  df-swapf 49918

This theorem is referenced by:  swapfcoa  49939