Theorem uspgrbisymrelALT 48130

Description: Alternate proof of uspgrbisymrel 48129 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 6889	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
2	1	pwex 5350	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
3		mptexg 7213	. . . 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 2735	. . . . 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 48125	. . . 4 ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V)
8		mptexg 7213	. . . 4 ⊢ (𝐺 ∈ V → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
10		coexg 7925	. . 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 2735	. . . 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 47512	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)–1-1-onto→𝑅)
15		eqid 2735	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
16	5, 6, 15	uspgrsprf1o 48124	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺–1-1-onto→𝒫 (Pairs‘𝑉))
17		f1oco 6841	. . 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 6806	. . 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 3576	. 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 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459  𝒫 cpw 4575  {cpr 4603   class class class wbr 5119  {copab 5181   ↦ cmpt 5201   × cxp 5652   ∘ ccom 5658  –1-1-onto→wf1o 6530  ‘cfv 6531  2^nd c2nd 7987  Vtxcvtx 28975  Edgcedg 29026  USPGraphcuspgr 29127  Pairscspr 47491

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-n0 12502  df-xnn0 12575  df-z 12589  df-uz 12853  df-fz 13525  df-hash 14349  df-vtx 28977  df-iedg 28978  df-edg 29027  df-upgr 29061  df-uspgr 29129  df-spr 47492

This theorem is referenced by: (None)