Theorem uspgrbisymrelALT 45205

Description: Alternate proof of uspgrbisymrel 45204 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 6769	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
2	1	pwex 5298	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
3		mptexg 7079	. . . 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 2738	. . . . 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 45200	. . . 4 ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V)
8		mptexg 7079	. . . 4 ⊢ (𝐺 ∈ V → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
10		coexg 7750	. . 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 583	. 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 2738	. . . 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 44838	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)–1-1-onto→𝑅)
15		eqid 2738	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
16	5, 6, 15	uspgrsprf1o 45199	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺–1-1-onto→𝒫 (Pairs‘𝑉))
17		f1oco 6722	. . 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 583	. 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 6688	. . 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 3526	. 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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422  𝒫 cpw 4530  {cpr 4560   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   × cxp 5578   ∘ ccom 5584  –1-1-onto→wf1o 6417  ‘cfv 6418  2^nd c2nd 7803  Vtxcvtx 27269  Edgcedg 27320  USPGraphcuspgr 27421  Pairscspr 44817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-vtx 27271  df-iedg 27272  df-edg 27321  df-upgr 27355  df-uspgr 27423  df-spr 44818

This theorem is referenced by: (None)