Theorem swapf2f1oaALT 50089

Description: Alternate proof of swapf2f1oa 50088. (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 2765	. . 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 9057	. 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 50081	. . . 4 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
11		eqid 2765	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2765	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13	4, 5, 11, 12, 8, 6, 7	xpchom 18254	. . . . 5 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
14	13	mpteq1d 5203	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}) = (𝑓 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) ↦ ∪ ◡{𝑓}))
15	10, 14	eqtrd 2800	. . 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 2765	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
18		swapf2f1o.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝑇)
19	4, 5, 6	elxpcbasex1 50059	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20	4, 5, 6	elxpcbasex2 50061	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
21	3, 4, 16, 19, 20, 5, 17	swapf1f1o 50086	. . . . . . 7 ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→(Base‘𝑇))
22		f1of 6824	. . . . . . 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 7084	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘𝑇))
25	23, 7	ffvelcdmd 7084	. . . . 5 ⊢ (𝜑 → (𝑂‘𝑌) ∈ (Base‘𝑇))
26	16, 17, 12, 11, 18, 24, 25	xpchom 18254	. . . 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 50080	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
28	27	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
29		fvex 6898	. . . . . . . 8 ⊢ (2nd ‘𝑋) ∈ V
30		fvex 6898	. . . . . . . 8 ⊢ (1st ‘𝑋) ∈ V
31	29, 30	op1st 7996	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (2nd ‘𝑋)
32	28, 31	eqtrdi 2816	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑋)) = (2nd ‘𝑋))
33	3, 4, 5, 7	swapf1a 50080	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)
34	33	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
35		fvex 6898	. . . . . . . 8 ⊢ (2nd ‘𝑌) ∈ V
36		fvex 6898	. . . . . . . 8 ⊢ (1st ‘𝑌) ∈ V
37	35, 36	op1st 7996	. . . . . . 7 ⊢ (1st ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (2nd ‘𝑌)
38	34, 37	eqtrdi 2816	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂‘𝑌)) = (2nd ‘𝑌))
39	32, 38	oveq12d 7434	. . . . 5 ⊢ (𝜑 → ((1st ‘(𝑂‘𝑋))(Hom ‘𝐷)(1st ‘(𝑂‘𝑌))) = ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)))
40	27	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩))
41	29, 30	op2nd 7997	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑋), (1st ‘𝑋)⟩) = (1st ‘𝑋)
42	40, 41	eqtrdi 2816	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑋)) = (1st ‘𝑋))
43	33	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
44	35, 36	op2nd 7997	. . . . . . 7 ⊢ (2nd ‘⟨(2nd ‘𝑌), (1st ‘𝑌)⟩) = (1st ‘𝑌)
45	43, 44	eqtrdi 2816	. . . . . 6 ⊢ (𝜑 → (2nd ‘(𝑂‘𝑌)) = (1st ‘𝑌))
46	42, 45	oveq12d 7434	. . . . 5 ⊢ (𝜑 → ((2nd ‘(𝑂‘𝑋))(Hom ‘𝐶)(2nd ‘(𝑂‘𝑌))) = ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)))
47	39, 46	xpeq12d 5694	. . . 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 2800	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌)) × ((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌))))
49	15, 13, 48	f1oeq123d 6818	. 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)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  ⟨cop 4597  ∪ cuni 4874   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17287  Hom chom 17339   ×_c cxpc 18242   swap_F cswapf 50070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-struct 17225  df-slot 17260  df-ndx 17272  df-base 17288  df-hom 17352  df-cco 17353  df-xpc 18246  df-swapf 50071

This theorem is used by: (None)