Theorem uspgrbisymrelALT 49526

Description: Alternate proof of uspgrbisymrel 49525 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⟶1-1onto𝑅)

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 6929	. . . . 5 ⊢ (Pairs‘𝑉) ∈ V
2	1	pwex 5349	. . . 4 ⊢ 𝒫 (Pairs‘𝑉) ∈ V
3		mptexg 7258	. . . 4 ⊢ (𝒫 (Pairs‘𝑉) ∈ V → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
4	2, 3	mp1i 14	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∈ V)
5		eqid 2763	. . . . 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 49521	. . . 4 ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V)
8		mptexg 7258	. . . 4 ⊢ (𝐺 ∈ V → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
9	7, 8	syl 18	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ∈ V)
10		coexg 7975	. . 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 596	. 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 2763	. . . 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 48853	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)⟶1-1onto𝑅)
15		eqid 2763	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
16	5, 6, 15	uspgrsprf1o 49520	. . 3 ⊢ (𝑉 ∈ 𝑊 → (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺⟶1-1onto𝒫 (Pairs‘𝑉))
17		f1oco 6878	. . 3 ⊢ (((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}):𝒫 (Pairs‘𝑉)⟶1-1onto𝑅 ∧ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)):𝐺⟶1-1onto𝒫 (Pairs‘𝑉)) → ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺⟶1-1onto𝑅)
18	14, 16, 17	syl2anc 596	. 2 ⊢ (𝑉 ∈ 𝑊 → ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺⟶1-1onto𝑅)
19		f1oeq1 6842	. . 3 ⊢ (𝑓 = ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))) → (𝑓:𝐺⟶1-1onto𝑅 ↔ ((𝑝 ∈ 𝒫 (Pairs‘𝑉) ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ∘ (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))):𝐺⟶1-1onto𝑅))
20	19	spcegv 3555	. 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⟶1-1onto𝑅 → ∃𝑓 𝑓:𝐺⟶1-1onto𝑅))
21	11, 18, 20	sylc 66	1 ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺⟶1-1onto𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  𝒫 cpw 4560  {cpr 4590   class class class wbr 5108  {copab 5173   ↦ cmpt 5192   × cxp 5656   ∘ ccom 5662  ⟶1-1ontowf1o 6565  ‘cfv 6566  2^nd c2nd 8034  Vtxcvtx 29631  Edgcedg 29682  USPGraphcuspgr 29786  Pairscspr 48832

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-fz 13680  df-hash 14516  df-vtx 29633  df-iedg 29634  df-edg 29683  df-upgr 29717  df-uspgr 29788  df-spr 48833

This theorem is used by: (None)