Theorem vtxvalg 15940

Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

Assertion

Ref	Expression
vtxvalg	⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)))

Proof of Theorem vtxvalg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vtx 15938	. 2 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
2		eleq1 2294	. . 3 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3		fveq2 5648	. . 3 ⊢ (𝑔 = 𝐺 → (1st ‘𝑔) = (1st ‘𝐺))
4		fveq2 5648	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5	2, 3, 4	ifbieq12d 3636	. 2 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)))
6		elex 2815	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		1stexg 6339	. . 3 ⊢ (𝐺 ∈ 𝑉 → (1st ‘𝐺) ∈ V)
8		basfn 13204	. . . 4 ⊢ Base Fn V
9		funfvex 5665	. . . . 5 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
10	9	funfni 5439	. . . 4 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
11	8, 6, 10	sylancr 414	. . 3 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
12	7, 11	ifexd 4587	. 2 ⊢ (𝐺 ∈ 𝑉 → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) ∈ V)
13	1, 5, 6, 12	fvmptd3 5749	1 ⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803  ifcif 3607   × cxp 4729   Fn wfn 5328  ‘cfv 5333  1^st c1st 6310  Basecbs 13145  Vtxcvtx 15936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-vtx 15938

This theorem is referenced by:  vtxex  15942  opvtxval  15945  funvtxdm2domval  15953  funvtxdm2vald  15955  vtxval0  15977