Theorem uspgredg2v 16101

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 16100	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
5	4	ralrimiva 2604	. . 3 ⊢ (𝐺 ∈ USPGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
6	5	adantr 276	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7		preq2 3750	. . . . . . 7 ⊢ (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
8	7	eqeq2d 2242	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
9	8	cbvriotavw 5987	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
10	7	eqeq2d 2242	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
11	10	cbvriotavw 5987	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
12		simpl 109	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USPGraph)
13		eleq2w 2292	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑦 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑦))
14	13, 3	elrab2 2964	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦))
15	2	eleq2i 2297	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐸 ↔ 𝑦 ∈ (Edg‘𝐺))
16	15	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ (Edg‘𝐺))
17	16	anim1i 340	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
18	14, 17	sylbi 121	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
19	18	adantr 276	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
20	12, 19	anim12i 338	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
21		3anass 1008	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
22	20, 21	sylibr 134	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
23		uspgredg2vtxeu 16098	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24		reueq1 2731	. . . . . . . 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 134	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) → ∃!𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
27	22, 26	syl 14	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃!𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
28		eleq2w 2292	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑥 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑥))
29	28, 3	elrab2 2964	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥))
30	2	eleq2i 2297	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (Edg‘𝐺))
31	30	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ (Edg‘𝐺))
32	31	anim1i 340	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
33	29, 32	sylbi 121	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
34	33	adantl 277	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
35	12, 34	anim12i 338	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
36		3anass 1008	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
37	35, 36	sylibr 134	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
38		uspgredg2vtxeu 16098	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39		reueq1 2731	. . . . . . . 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 134	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) → ∃!𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
42	37, 41	syl 14	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ∃!𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
43	9, 11, 27, 42	riotaeqimp 6001	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
44	43	ex 115	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
45	44	ralrimivva 2613	. 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 2237	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
48	47	riotabidv 5978	. . 3 ⊢ (𝑦 = 𝑥 → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}))
49	46, 48	f1mpt 5917	. 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 417	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201  ∀wral 2509  ∃!wreu 2511  {crab 2513  {cpr 3671   ↦ cmpt 4151  –1-1→wf1 5325  ‘cfv 5328  ℩crio 5975  Vtxcvtx 15892  Edgcedg 15937  USPGraphcuspgr 16033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-1o 6587  df-2o 6588  df-en 6915  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-edg 15938  df-upgren 15973  df-uspgren 16035

This theorem is referenced by:  uspgredgdomord  16109