Theorem uspgredg2v 29312

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 29311	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
5	4	ralrimiva 3131	. . 3 ⊢ (𝐺 ∈ USPGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
6	5	adantr 481	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7		preq2 4667	. . . . . . 7 ⊢ (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
8	7	eqeq2d 2750	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
9	8	cbvriotavw 7324	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
10	7	eqeq2d 2750	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
11	10	cbvriotavw 7324	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
12		simpl 483	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USPGraph)
13		eleq2w 2823	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑦 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑦))
14	13, 3	elrab2 3632	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦))
15	2	eleq2i 2831	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐸 ↔ 𝑦 ∈ (Edg‘𝐺))
16	15	biimpi 217	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ (Edg‘𝐺))
17	16	anim1i 621	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
18	14, 17	sylbi 218	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
19	18	adantr 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
20	12, 19	anim12i 619	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
21		3anass 1100	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
22	20, 21	sylibr 235	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
23		uspgredg2vtxeu 29308	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24		reueq1 3376	. . . . . . . 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 235	. . . . . 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 2823	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑥 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑥))
29	28, 3	elrab2 3632	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥))
30	2	eleq2i 2831	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (Edg‘𝐺))
31	30	biimpi 217	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ (Edg‘𝐺))
32	31	anim1i 621	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
33	29, 32	sylbi 218	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
34	33	adantl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
35	12, 34	anim12i 619	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
36		3anass 1100	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
37	35, 36	sylibr 235	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
38		uspgredg2vtxeu 29308	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39		reueq1 3376	. . . . . . . 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 235	. . . . . 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 7340	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
44	43	ex 413	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
45	44	ralrimivva 3182	. 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 2743	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
48	47	riotabidv 7316	. . 3 ⊢ (𝑦 = 𝑥 → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}))
49	46, 48	f1mpt 7206	. 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 589	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃!wreu 3342  {crab 3391  {cpr 4558   ↦ cmpt 5154  –1-1→wf1 6483  ‘cfv 6486  ℩crio 7313  Vtxcvtx 29084  Edgcedg 29135  USPGraphcuspgr 29236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-oadd 8400  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-dju 9817  df-card 9855  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-n0 12430  df-xnn0 12503  df-z 12517  df-uz 12781  df-fz 13454  df-hash 14285  df-edg 29136  df-upgr 29170  df-uspgr 29238

This theorem is referenced by:  uspgredgleord  29320