Theorem opvtxval 27662

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

Assertion

Ref	Expression
opvtxval	⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺))

Proof of Theorem opvtxval

Step	Hyp	Ref	Expression
1		vtxval 27659	. 2 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))
2		iftrue 4484	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (1st ‘𝐺))
3	1, 2	eqtrid 2789	1 ⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3442  ifcif 4478   × cxp 5623  ‘cfv 6484  1^st c1st 7902  Basecbs 17010  Vtxcvtx 27655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6436  df-fun 6486  df-fv 6492  df-vtx 27657

This theorem is referenced by:  opvtxfv  27663