Theorem uspgrbisymrelALT 43432

Description: Alternate proof of uspgrbisymrel 43431 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 6509	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
2	1	pwex 5130	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
3		mptexg 6808	. . . 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 2771	. . . . 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 43427	. . . 4 ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V)
8		mptexg 6808	. . . 4 ⊢ (𝐺 ∈ V → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
10		coexg 7447	. . 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 576	. 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 2771	. . . 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 43062	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)–1-1-onto→𝑅)
15		eqid 2771	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
16	5, 6, 15	uspgrsprf1o 43426	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺–1-1-onto→𝒫 (Pairs‘𝑉))
17		f1oco 6463	. . 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 576	. 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 6430	. . 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 3509	. 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 198   ∧ wa 387   = wceq 1508  ∃wex 1743   ∈ wcel 2051  ∀wral 3081  ∃wrex 3082  {crab 3085  Vcvv 3408  𝒫 cpw 4416  {cpr 4437   class class class wbr 4925  {copab 4987   ↦ cmpt 5004   × cxp 5401   ∘ ccom 5407  –1-1-onto→wf1o 6184  ‘cfv 6185  2^nd c2nd 7498  Vtxcvtx 26499  Edgcedg 26550  USPGraphcuspgr 26651  Pairscspr 43041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-dju 9122  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-2 11501  df-n0 11706  df-xnn0 11778  df-z 11792  df-uz 12057  df-fz 12707  df-hash 13504  df-vtx 26501  df-iedg 26502  df-edg 26551  df-upgr 26585  df-uspgr 26653  df-spr 43042

This theorem is referenced by: (None)