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Theorem cplgruvtxbOLD 26545
Description: Obsolete proof of cplgruvtxb 26542 as of 15-Feb-2022. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
cplgruvtxb.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
cplgruvtxbOLD (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Proof of Theorem cplgruvtxbOLD
Dummy variables 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cplgruvtxb.v . . 3 𝑉 = (Vtx‘𝐺)
21iscplgr 26544 . 2 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
31uvtxisvtx 26515 . . . . . . . . 9 (𝑔 ∈ (UnivVtx‘𝐺) → 𝑔𝑉)
43adantl 467 . . . . . . . 8 ((𝐺𝑊𝑔 ∈ (UnivVtx‘𝐺)) → 𝑔𝑉)
54ralrimiva 3114 . . . . . . 7 (𝐺𝑊 → ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔𝑉)
6 dfss3 3739 . . . . . . 7 ((UnivVtx‘𝐺) ⊆ 𝑉 ↔ ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔𝑉)
75, 6sylibr 224 . . . . . 6 (𝐺𝑊 → (UnivVtx‘𝐺) ⊆ 𝑉)
87adantr 466 . . . . 5 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) ⊆ 𝑉)
9 dfss3 3739 . . . . . . 7 (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
109biimpri 218 . . . . . 6 (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) → 𝑉 ⊆ (UnivVtx‘𝐺))
1110adantl 467 . . . . 5 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → 𝑉 ⊆ (UnivVtx‘𝐺))
128, 11eqssd 3767 . . . 4 ((𝐺𝑊 ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) = 𝑉)
1312ex 397 . . 3 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) → (UnivVtx‘𝐺) = 𝑉))
14 raleleq 3304 . . . 4 (𝑉 = (UnivVtx‘𝐺) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
1514eqcoms 2778 . . 3 ((UnivVtx‘𝐺) = 𝑉 → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
1613, 15impbid1 215 . 2 (𝐺𝑊 → (∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ (UnivVtx‘𝐺) = 𝑉))
172, 16bitrd 268 1 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wss 3721  cfv 6031  Vtxcvtx 26094  UnivVtxcuvtx 26509  ComplGraphccplgr 26538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-uvtx 26510  df-cplgr 26540
This theorem is referenced by: (None)
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