Theorem swapf2a 50455

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 50454	. 2 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
8		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9	8	sneqd 4595	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → {𝑓} = {𝐹})
10	9	cnveqd 5852	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡{𝑓} = ◡{𝐹})
11	10	unieqd 4879	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ∪ ◡{𝐹})
12		swapf2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
13	6	oveqd 7437	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝑆)𝑌))
14		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
17	2, 3, 14, 15, 16, 4, 5	xpchom 18355	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝑆)𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
18	13, 17	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
19	12, 18	eleqtrd 2862	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
20		2nd1st 8049	. . . . 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 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
23	11, 22	eqtrd 2795	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
24		opex 5431	. . 3 ⊢ ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V)
26	7, 23, 12, 25	fvmptd 7001	1 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4583  ⟨cop 4589  ∪ cuni 4866   × cxp 5648  ^◡ccnv 5649  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17388  Hom chom 17440   ×_c cxpc 18343   swap_F cswapf 50443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-er 8718  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-3 12407  df-4 12408  df-5 12409  df-6 12410  df-7 12411  df-8 12412  df-9 12413  df-n0 12608  df-z 12695  df-dec 12816  df-uz 12967  df-fz 13641  df-struct 17326  df-slot 17361  df-ndx 17373  df-base 17389  df-hom 17453  df-cco 17454  df-xpc 18347  df-swapf 50444

This theorem is used by:  swapfcoa  50465