Theorem uspgrbisymrelALT 46519

Description: Alternate proof of uspgrbisymrel 46518 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
uspgrbisymrel.g	⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
uspgrbisymrel.r	⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}

Assertion

Ref	Expression
uspgrbisymrelALT	⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅)

Distinct variable groups:   𝑒,𝑉,𝑞,𝑣   𝑉,𝑟,𝑥,𝑦   𝑒,𝑊,𝑞,𝑣   𝑥,𝑊,𝑦   𝑓,𝐺   𝑅,𝑓   𝑓,𝑉,𝑟,𝑥,𝑦

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑣,𝑒,𝑟,𝑞)   𝐺(𝑥,𝑦,𝑣,𝑒,𝑟,𝑞)   𝑊(𝑓,𝑟)

Proof of Theorem uspgrbisymrelALT

Dummy variables 𝑔 𝑝 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6901	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
2	1	pwex 5377	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
3		mptexg 7219	. . . 4 ⊢ (𝒫 (Pairs‘𝑉) ∈ V → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
4	2, 3	mp1i 13	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
5		eqid 2732	. . . . 5 ⊢ 𝒫 (Pairs‘𝑉) = 𝒫 (Pairs‘𝑉)
6		uspgrbisymrel.g	. . . . 5 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
7	5, 6	uspgrex 46514	. . . 4 ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V)
8		mptexg 7219	. . . 4 ⊢ (𝐺 ∈ V → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
10		coexg 7916	. . 3 ⊢ (((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V ∧ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V) → ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))) ∈ V)
11	4, 9, 10	syl2anc 584	. 2 ⊢ (𝑉 ∈ 𝑊 → ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))) ∈ V)
12		uspgrbisymrel.r	. . . 4 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
13		eqid 2732	. . . 4 ⊢ (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) = (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
14	5, 12, 13	sprsymrelf1o 46152	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)–1-1-onto→𝑅)
15		eqid 2732	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
16	5, 6, 15	uspgrsprf1o 46513	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺–1-1-onto→𝒫 (Pairs‘𝑉))
17		f1oco 6853	. . 3 ⊢ (((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)–1-1-onto→𝑅 ∧ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺–1-1-onto→𝒫 (Pairs‘𝑉)) → ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺–1-1-onto→𝑅)
18	14, 16, 17	syl2anc 584	. 2 ⊢ (𝑉 ∈ 𝑊 → ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺–1-1-onto→𝑅)
19		f1oeq1 6818	. . 3 ⊢ (𝑓 = ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))) → (𝑓:𝐺–1-1-onto→𝑅 ↔ ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺–1-1-onto→𝑅))
20	19	spcegv 3587	. 2 ⊢ (((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))) ∈ V → (((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺–1-1-onto→𝑅 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅))
21	11, 18, 20	sylc 65	1 ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474  𝒫 cpw 4601  {cpr 4629   class class class wbr 5147  {copab 5209   ↦ cmpt 5230   × cxp 5673   ∘ ccom 5679  –1-1-onto→wf1o 6539  ‘cfv 6540  2^nd c2nd 7970  Vtxcvtx 28245  Edgcedg 28296  USPGraphcuspgr 28397  Pairscspr 46131

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-vtx 28247  df-iedg 28248  df-edg 28297  df-upgr 28331  df-uspgr 28399  df-spr 46132

This theorem is referenced by: (None)