Theorem swapf2f1oaALT 49249

Description: Alternate proof of swapf2f1oa 49248. (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 2730	. . 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 9034	. 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 49241	. . . 4 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
11		eqid 2730	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2730	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13	4, 5, 11, 12, 8, 6, 7	xpchom 18147	. . . . 5 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
14	13	mpteq1d 5199	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}))
15	10, 14	eqtrd 2765	. . 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 2730	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
18		swapf2f1o.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝑇)
19	4, 5, 6	elxpcbasex1 49219	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20	4, 5, 6	elxpcbasex2 49221	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
21	3, 4, 16, 19, 20, 5, 17	swapf1f1o 49246	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→(Base‘𝑇))
22		f1of 6802	. . . . . . 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 7059	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘𝑇))
25	23, 7	ffvelcdmd 7059	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑌) ∈ (Base‘𝑇))
26	16, 17, 12, 11, 18, 24, 25	xpchom 18147	. . . 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 49240	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
28	27	fveq2d 6864	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
29		fvex 6873	. . . . . . . 8 ⊢ (2nd ‘𝑋) ∈ V
30		fvex 6873	. . . . . . . 8 ⊢ (1st ‘𝑋) ∈ V
31	29, 30	op1st 7978	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (2nd ‘𝑋)
32	28, 31	eqtrdi 2781	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (2nd ‘𝑋))
33	3, 4, 5, 7	swapf1a 49240	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)
34	33	fveq2d 6864	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
35		fvex 6873	. . . . . . . 8 ⊢ (2nd ‘𝑌) ∈ V
36		fvex 6873	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
37	35, 36	op1st 7978	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (2nd ‘𝑌)
38	34, 37	eqtrdi 2781	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (2nd ‘𝑌))
39	32, 38	oveq12d 7407	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) = ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))
40	27	fveq2d 6864	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
41	29, 30	op2nd 7979	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (1st ‘𝑋)
42	40, 41	eqtrdi 2781	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (1st ‘𝑋))
43	33	fveq2d 6864	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
44	35, 36	op2nd 7979	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (1st ‘𝑌)
45	43, 44	eqtrdi 2781	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (1st ‘𝑌))
46	42, 45	oveq12d 7407	. . . . 5 ⊢ (𝜑 → ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌))) = ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))
47	39, 46	xpeq12d 5671	. . . 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 2765	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌))))
49	15, 13, 48	f1oeq123d 6796	. 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 3450  {csn 4591  ⟨cop 4597  ∪ cuni 4873   ↦ cmpt 5190   × cxp 5638  ^◡ccnv 5639  ⟶wf 6509  –1-1-onto→wf1o 6512  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  Basecbs 17185  Hom chom 17237   ×_c cxpc 18135   swap_F cswapf 49230

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-xpc 18139  df-swapf 49231

This theorem is referenced by: (None)