Theorem swapf2f1oaALT 49283

Description: Alternate proof of swapf2f1oa 49282. (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 2729	. . 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 8990	. 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 49275	. . . 4 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
11		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13	4, 5, 11, 12, 8, 6, 7	xpchom 18105	. . . . 5 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
14	13	mpteq1d 5185	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}))
15	10, 14	eqtrd 2764	. . 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 2729	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
18		swapf2f1o.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝑇)
19	4, 5, 6	elxpcbasex1 49253	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20	4, 5, 6	elxpcbasex2 49255	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
21	3, 4, 16, 19, 20, 5, 17	swapf1f1o 49280	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→(Base‘𝑇))
22		f1of 6768	. . . . . . 7 ⊢ (𝑂:𝐵–1-1-onto→(Base‘𝑇) → 𝑂:𝐵⟶(Base‘𝑇))
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘𝑇))
24	23, 6	ffvelcdmd 7023	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘𝑇))
25	23, 7	ffvelcdmd 7023	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑌) ∈ (Base‘𝑇))
26	16, 17, 12, 11, 18, 24, 25	xpchom 18105	. . . 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 49274	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
28	27	fveq2d 6830	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
29		fvex 6839	. . . . . . . 8 ⊢ (2nd ‘𝑋) ∈ V
30		fvex 6839	. . . . . . . 8 ⊢ (1st ‘𝑋) ∈ V
31	29, 30	op1st 7939	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (2nd ‘𝑋)
32	28, 31	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (2nd ‘𝑋))
33	3, 4, 5, 7	swapf1a 49274	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)
34	33	fveq2d 6830	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
35		fvex 6839	. . . . . . . 8 ⊢ (2nd ‘𝑌) ∈ V
36		fvex 6839	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
37	35, 36	op1st 7939	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (2nd ‘𝑌)
38	34, 37	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (2nd ‘𝑌))
39	32, 38	oveq12d 7371	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) = ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))
40	27	fveq2d 6830	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
41	29, 30	op2nd 7940	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (1st ‘𝑋)
42	40, 41	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (1st ‘𝑋))
43	33	fveq2d 6830	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
44	35, 36	op2nd 7940	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (1st ‘𝑌)
45	43, 44	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (1st ‘𝑌))
46	42, 45	oveq12d 7371	. . . . 5 ⊢ (𝜑 → ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌))) = ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))
47	39, 46	xpeq12d 5654	. . . 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 2764	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌))))
49	15, 13, 48	f1oeq123d 6762	. 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 258	1 ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579  ⟨cop 4585  ∪ cuni 4861   ↦ cmpt 5176   × cxp 5621  ^◡ccnv 5622  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930  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-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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-nel 3030  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-riota 7310  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-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12611  df-uz 12755  df-fz 13430  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17140  df-hom 17204  df-cco 17205  df-xpc 18097  df-swapf 49265

This theorem is referenced by: (None)