Theorem opvtxval 29293

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 29290	. 2 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))
2		iftrue 4498	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (1st ‘𝐺))
3	1, 2	eqtrid 2816	1 ⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ifcif 4492   × cxp 5660  ‘cfv 6537  1^st c1st 7983  Basecbs 17268  Vtxcvtx 29286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-vtx 29288

This theorem is referenced by:  opvtxfv  29294