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	⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Proof of Theorem cplgruvtxbOLD

Dummy variables 𝑔 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cplgruvtxb.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	iscplgr 26544	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
3	1	uvtxisvtx 26515	. . . . . . . . 9 ⊢ (𝑔 ∈ (UnivVtx‘𝐺) → 𝑔 ∈ 𝑉)
4	3	adantl 467	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑔 ∈ (UnivVtx‘𝐺)) → 𝑔 ∈ 𝑉)
5	4	ralrimiva 3114	. . . . . . 7 ⊢ (𝐺 ∈ 𝑊 → ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔 ∈ 𝑉)
6		dfss3 3739	. . . . . . 7 ⊢ ((UnivVtx‘𝐺) ⊆ 𝑉 ↔ ∀𝑔 ∈ (UnivVtx‘𝐺)𝑔 ∈ 𝑉)
7	5, 6	sylibr 224	. . . . . 6 ⊢ (𝐺 ∈ 𝑊 → (UnivVtx‘𝐺) ⊆ 𝑉)
8	7	adantr 466	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) ⊆ 𝑉)
9		dfss3 3739	. . . . . . 7 ⊢ (𝑉 ⊆ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
10	9	biimpri 218	. . . . . 6 ⊢ (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) → 𝑉 ⊆ (UnivVtx‘𝐺))
11	10	adantl 467	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → 𝑉 ⊆ (UnivVtx‘𝐺))
12	8, 11	eqssd 3767	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) → (UnivVtx‘𝐺) = 𝑉)
13	12	ex 397	. . 3 ⊢ (𝐺 ∈ 𝑊 → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) → (UnivVtx‘𝐺) = 𝑉))
14		raleleq 3304	. . . 4 ⊢ (𝑉 = (UnivVtx‘𝐺) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
15	14	eqcoms 2778	. . 3 ⊢ ((UnivVtx‘𝐺) = 𝑉 → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
16	13, 15	impbid1 215	. 2 ⊢ (𝐺 ∈ 𝑊 → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ (UnivVtx‘𝐺) = 𝑉))
17	2, 16	bitrd 268	1 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

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)