Theorem uspgredg2v 28914

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 28913	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
5	4	ralrimiva 3145	. . 3 ⊢ (𝐺 ∈ USPGraph → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
6	5	adantr 480	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7		preq2 4738	. . . . . . 7 ⊢ (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
8	7	eqeq2d 2742	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
9	8	cbvriotavw 7378	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑦 = {𝑁, 𝑛})
10	7	eqeq2d 2742	. . . . . 6 ⊢ (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
11	10	cbvriotavw 7378	. . . . 5 ⊢ (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) = (℩𝑛 ∈ 𝑉 𝑥 = {𝑁, 𝑛})
12		simpl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐺 ∈ USPGraph)
13		eleq2w 2816	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑦 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑦))
14	13, 3	elrab2 3686	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦))
15	2	eleq2i 2824	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐸 ↔ 𝑦 ∈ (Edg‘𝐺))
16	15	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐸 → 𝑦 ∈ (Edg‘𝐺))
17	16	anim1i 614	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐸 ∧ 𝑁 ∈ 𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
18	14, 17	sylbi 216	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
20	12, 19	anim12i 612	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
21		3anass 1094	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦)))
22	20, 21	sylibr 233	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦))
23		uspgredg2vtxeu 28910	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24		reueq1 3414	. . . . . . . 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 233	. . . . . 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 2816	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑥 → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑥))
29	28, 3	elrab2 3686	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥))
30	2	eleq2i 2824	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐸 ↔ 𝑥 ∈ (Edg‘𝐺))
31	30	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐸 → 𝑥 ∈ (Edg‘𝐺))
32	31	anim1i 614	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐸 ∧ 𝑁 ∈ 𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
33	29, 32	sylbi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
34	33	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
35	12, 34	anim12i 612	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
36		3anass 1094	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥)))
37	35, 36	sylibr 233	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥))
38		uspgredg2vtxeu 28910	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁 ∈ 𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39		reueq1 3414	. . . . . . . 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 233	. . . . . 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 7395	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
44	43	ex 412	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → ((℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
45	44	ralrimivva 3199	. 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 2735	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
48	47	riotabidv 7370	. . 3 ⊢ (𝑦 = 𝑥 → (℩𝑧 ∈ 𝑉 𝑦 = {𝑁, 𝑧}) = (℩𝑧 ∈ 𝑉 𝑥 = {𝑁, 𝑧}))
49	46, 48	f1mpt 7263	. 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 582	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃!wreu 3373  {crab 3431  {cpr 4630   ↦ cmpt 5231  –1-1→wf1 6540  ‘cfv 6543  ℩crio 7367  Vtxcvtx 28689  Edgcedg 28740  USPGraphcuspgr 28841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-oadd 8476  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-fz 13492  df-hash 14298  df-edg 28741  df-upgr 28775  df-uspgr 28843

This theorem is referenced by:  uspgredgleord  28922