Theorem opvtxval 29514

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 29511	. 2 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))
2		iftrue 4487	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (1st ‘𝐺))
3	1, 2	eqtrid 2807	1 ⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ifcif 4481   × cxp 5645  ‘cfv 6527  1^st c1st 7982  Basecbs 17348  Vtxcvtx 29507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6483  df-fun 6529  df-fv 6535  df-vtx 29509

This theorem is used by:  opvtxfv  29515