Theorem swapf2f1oaALT 49936

Description: Alternate proof of swapf2f1oa 49935. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
swapf1f1o.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
swapf1f1o.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf1f1o.t	⊢ 𝑇 = (𝐷 ×c 𝐶)
swapf2f1o.h	⊢ 𝐻 = (Hom ‘𝑆)
swapf2f1o.j	⊢ 𝐽 = (Hom ‘𝑇)
swapf2f1oa.b	⊢ 𝐵 = (Base‘𝑆)
swapf2f1oa.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapf2f1oa.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
swapf2f1oaALT	⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

Proof of Theorem swapf2f1oaALT

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓})
2	1	xpcomf1o 9050	. 2 ⊢ (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}):(((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))–1-1-onto→(((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))
3		swapf1f1o.o	. . . . 5 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
4		swapf1f1o.s	. . . . 5 ⊢ 𝑆 = (𝐶 ×c 𝐷)
5		swapf2f1oa.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
6		swapf2f1oa.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		swapf2f1oa.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		swapf2f1o.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝑆)
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
10	3, 4, 5, 6, 7, 9	swapf2vala 49928	. . . 4 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
11		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13	4, 5, 11, 12, 8, 6, 7	xpchom 18232	. . . . 5 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
14	13	mpteq1d 5202	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}))
15	10, 14	eqtrd 2804	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}))
16		swapf1f1o.t	. . . . 5 ⊢ 𝑇 = (𝐷 ×c 𝐶)
17		eqid 2769	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
18		swapf2f1o.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝑇)
19	4, 5, 6	elxpcbasex1 49906	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20	4, 5, 6	elxpcbasex2 49908	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
21	3, 4, 16, 19, 20, 5, 17	swapf1f1o 49933	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→(Base‘𝑇))
22		f1of 6818	. . . . . . 7 ⊢ (𝑂:𝐵–1-1-onto→(Base‘𝑇) → 𝑂:𝐵⟶(Base‘𝑇))
23	21, 22	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘𝑇))
24	23, 6	ffvelcdmd 7078	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘𝑇))
25	23, 7	ffvelcdmd 7078	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑌) ∈ (Base‘𝑇))
26	16, 17, 12, 11, 18, 24, 25	xpchom 18232	. . . 4 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) × ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌)))))
27	3, 4, 5, 6	swapf1a 49927	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
28	27	fveq2d 6883	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
29		fvex 6892	. . . . . . . 8 ⊢ (2nd ‘𝑋) ∈ V
30		fvex 6892	. . . . . . . 8 ⊢ (1st ‘𝑋) ∈ V
31	29, 30	op1st 7990	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (2nd ‘𝑋)
32	28, 31	eqtrdi 2820	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (2nd ‘𝑋))
33	3, 4, 5, 7	swapf1a 49927	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)
34	33	fveq2d 6883	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
35		fvex 6892	. . . . . . . 8 ⊢ (2nd ‘𝑌) ∈ V
36		fvex 6892	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
37	35, 36	op1st 7990	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (2nd ‘𝑌)
38	34, 37	eqtrdi 2820	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (2nd ‘𝑌))
39	32, 38	oveq12d 7426	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) = ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))
40	27	fveq2d 6883	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
41	29, 30	op2nd 7991	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (1st ‘𝑋)
42	40, 41	eqtrdi 2820	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (1st ‘𝑋))
43	33	fveq2d 6883	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
44	35, 36	op2nd 7991	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (1st ‘𝑌)
45	43, 44	eqtrdi 2820	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (1st ‘𝑌))
46	42, 45	oveq12d 7426	. . . . 5 ⊢ (𝜑 → ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌))) = ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))
47	39, 46	xpeq12d 5690	. . . 4 ⊢ (𝜑 → (((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) × ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌)))) = (((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌))))
48	26, 47	eqtrd 2804	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌))))
49	15, 13, 48	f1oeq123d 6812	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)) ↔ (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}):(((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))–1-1-onto→(((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))))
50	2, 49	mpbiri 261	1 ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {csn 4591  ⟨cop 4597  ∪ cuni 4873   ↦ cmpt 5193   × cxp 5657  ^◡ccnv 5658  ⟶wf 6530  –1-1-onto→wf1o 6533  ‘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: (None)