Theorem swapf2f1oaALT 49753

Description: Alternate proof of swapf2f1oa 49752. (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 2736	. . 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 9004	. 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 49745	. . . 4 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
11		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13	4, 5, 11, 12, 8, 6, 7	xpchom 18146	. . . . 5 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
14	13	mpteq1d 5175	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}))
15	10, 14	eqtrd 2771	. . 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 2736	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
18		swapf2f1o.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝑇)
19	4, 5, 6	elxpcbasex1 49723	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20	4, 5, 6	elxpcbasex2 49725	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
21	3, 4, 16, 19, 20, 5, 17	swapf1f1o 49750	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→(Base‘𝑇))
22		f1of 6780	. . . . . . 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 7037	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘𝑇))
25	23, 7	ffvelcdmd 7037	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑌) ∈ (Base‘𝑇))
26	16, 17, 12, 11, 18, 24, 25	xpchom 18146	. . . 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 49744	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
28	27	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
29		fvex 6853	. . . . . . . 8 ⊢ (2nd ‘𝑋) ∈ V
30		fvex 6853	. . . . . . . 8 ⊢ (1st ‘𝑋) ∈ V
31	29, 30	op1st 7950	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (2nd ‘𝑋)
32	28, 31	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (2nd ‘𝑋))
33	3, 4, 5, 7	swapf1a 49744	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)
34	33	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
35		fvex 6853	. . . . . . . 8 ⊢ (2nd ‘𝑌) ∈ V
36		fvex 6853	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
37	35, 36	op1st 7950	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (2nd ‘𝑌)
38	34, 37	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (2nd ‘𝑌))
39	32, 38	oveq12d 7385	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) = ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))
40	27	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
41	29, 30	op2nd 7951	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (1st ‘𝑋)
42	40, 41	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (1st ‘𝑋))
43	33	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
44	35, 36	op2nd 7951	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (1st ‘𝑌)
45	43, 44	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (1st ‘𝑌))
46	42, 45	oveq12d 7385	. . . . 5 ⊢ (𝜑 → ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌))) = ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))
47	39, 46	xpeq12d 5662	. . . 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 2771	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌))))
49	15, 13, 48	f1oeq123d 6774	. 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 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567  ⟨cop 4573  ∪ cuni 4850   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Hom chom 17231   ×_c cxpc 18134   swap_F cswapf 49734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-xpc 18138  df-swapf 49735

This theorem is referenced by: (None)