Theorem uspgredg2v 27098

Description: In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)

Hypotheses

Ref	Expression
uspgredg2v.v	⊢ 𝑉 = (Vtx‘𝐺)
uspgredg2v.e	⊢ 𝐸 = (Edg‘𝐺)
uspgredg2v.a	⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
uspgredg2v.f	⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}))

Assertion

Ref	Expression
uspgredg2v	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Distinct variable groups:   𝑒,𝐸   𝑧,𝐺   𝑒,𝑁   𝑧,𝑁   𝑧,𝑉   𝑦,𝐴   𝑦,𝐺   𝑦,𝑁,𝑧   𝑦,𝑉   𝑦,𝑒

Allowed substitution hints:   𝐴(𝑧,𝑒)   𝐸(𝑦,𝑧)   𝐹(𝑦,𝑧,𝑒)   𝐺(𝑒)   𝑉(𝑒)

Proof of Theorem uspgredg2v

Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgredg2v.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		uspgredg2v.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3		uspgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}
4	1, 2, 3	uspgredg2vlem 27097	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
5	4	ralrimiva 3111	. . 3 ⊢ (𝐺 ∈ USPGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
6	5	adantr 485	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7		preq2 4620	. . . . . . 7 ⊢ (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
8	7	eqeq2d 2770	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
9	8	cbvriotavw 7111	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
10	7	eqeq2d 2770	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
11	10	cbvriotavw 7111	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
12		simpl 487	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USPGraph)
13		eleq2w 2834	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑦 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑦))
14	13, 3	elrab2 3603	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦))
15	2	eleq2i 2842	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐸 ↔ 𝑦 ∈ (Edg‘𝐺))
16	15	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ (Edg‘𝐺))
17	16	anim1i 618	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
18	14, 17	sylbi 220	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
19	18	adantr 485	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
20	12, 19	anim12i 616	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
21		3anass 1093	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
22	20, 21	sylibr 237	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
23		uspgredg2vtxeu 27094	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24		reueq1 3323	. . . . . . . 8 ⊢ (𝑉 = (Vtx‘𝐺) → (∃!𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛}))
25	1, 24	ax-mp 5	. . . . . . 7 ⊢ (∃!𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
26	23, 25	sylibr 237	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) → ∃!𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
27	22, 26	syl 17	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃!𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
28		eleq2w 2834	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑥 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑥))
29	28, 3	elrab2 3603	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥))
30	2	eleq2i 2842	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (Edg‘𝐺))
31	30	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ (Edg‘𝐺))
32	31	anim1i 618	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
33	29, 32	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
34	33	adantl 486	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
35	12, 34	anim12i 616	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
36		3anass 1093	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
37	35, 36	sylibr 237	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
38		uspgredg2vtxeu 27094	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39		reueq1 3323	. . . . . . . 8 ⊢ (𝑉 = (Vtx‘𝐺) → (∃!𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛}))
40	1, 39	ax-mp 5	. . . . . . 7 ⊢ (∃!𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
41	38, 40	sylibr 237	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) → ∃!𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
42	37, 41	syl 17	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃!𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
43	9, 11, 27, 42	riotaeqimp 7127	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
44	43	ex 417	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
45	44	ralrimivva 3118	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
46		uspgredg2v.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}))
47		eqeq1 2763	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
48	47	riotabidv 7103	. . 3 ⊢ (𝑦 = 𝑥 → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}))
49	46, 48	f1mpt 7004	. 2 ⊢ (𝐹:𝐴–1-1→𝑉 ↔ (∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥)))
50	6, 45, 49	sylanbrc 587	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3068  ∃!wreu 3070  {crab 3072  {cpr 4517   ↦ cmpt 5105  –1-1→wf1 6325  ‘cfv 6328  ℩crio 7100  Vtxcvtx 26873  Edgcedg 26924  USPGraphcuspgr 27025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-dju 9348  df-card 9386  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-2 11722  df-n0 11920  df-xnn0 11992  df-z 12006  df-uz 12268  df-fz 12925  df-hash 13726  df-edg 26925  df-upgr 26959  df-uspgr 27027

This theorem is referenced by:  uspgredgleord  27106