Theorem usgredg2vtxeuALT 29167

Description: Alternate proof of usgredg2vtxeu 29166, using edgiedgb 28999, the general translation from (iEdg‘𝐺) to (Edg‘𝐺). (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
usgredg2vtxeuALT	⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Distinct variable groups:   𝑦,𝐸   𝑦,𝐺   𝑦,𝑌

Proof of Theorem usgredg2vtxeuALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgruhgr 29131	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph)
2		eqid 2729	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	2	uhgredgiedgb 29071	. . . 4 ⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐸 = ((iEdg‘𝐺)‘𝑥)))
4	1, 3	syl 17	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐸 = ((iEdg‘𝐺)‘𝑥)))
5		eqid 2729	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
6	5, 2	usgredgreu 29163	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝑌 ∈ ((iEdg‘𝐺)‘𝑥)) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})
7	6	3expia 1121	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}))
8	7	3adant3 1132	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐸 = ((iEdg‘𝐺)‘𝑥)) → (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}))
9		eleq2 2817	. . . . . . . 8 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝑌 ∈ 𝐸 ↔ 𝑌 ∈ ((iEdg‘𝐺)‘𝑥)))
10		eqeq1 2733	. . . . . . . . 9 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝐸 = {𝑌, 𝑦} ↔ ((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}))
11	10	reubidv 3361	. . . . . . . 8 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → (∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦} ↔ ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦}))
12	9, 11	imbi12d 344	. . . . . . 7 ⊢ (𝐸 = ((iEdg‘𝐺)‘𝑥) → ((𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) ↔ (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})))
13	12	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐸 = ((iEdg‘𝐺)‘𝑥)) → ((𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}) ↔ (𝑌 ∈ ((iEdg‘𝐺)‘𝑥) → ∃!𝑦 ∈ (Vtx‘𝐺)((iEdg‘𝐺)‘𝑥) = {𝑌, 𝑦})))
14	8, 13	mpbird 257	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺) ∧ 𝐸 = ((iEdg‘𝐺)‘𝑥)) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}))
15	14	3exp 1119	. . . 4 ⊢ (𝐺 ∈ USGraph → (𝑥 ∈ dom (iEdg‘𝐺) → (𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦}))))
16	15	rexlimdv 3128	. . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑥 ∈ dom (iEdg‘𝐺)𝐸 = ((iEdg‘𝐺)‘𝑥) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})))
17	4, 16	sylbid 240	. 2 ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ (Edg‘𝐺) → (𝑌 ∈ 𝐸 → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})))
18	17	3imp 1110	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑌 ∈ 𝐸) → ∃!𝑦 ∈ (Vtx‘𝐺)𝐸 = {𝑌, 𝑦})

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ∃!wreu 3341  {cpr 4579  dom cdm 5619  ‘cfv 6482  Vtxcvtx 28941  iEdgciedg 28942  Edgcedg 28992  UHGraphcuhgr 29001  USGraphcusgr 29094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-hash 14238  df-edg 28993  df-uhgr 29003  df-upgr 29027  df-umgr 29028  df-uspgr 29095  df-usgr 29096

This theorem is referenced by: (None)