![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vtxval | Structured version Visualization version GIF version |
Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.) |
Ref | Expression |
---|---|
vtxval | ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2820 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V))) | |
2 | fveq2 6878 | . . . 4 ⊢ (𝑔 = 𝐺 → (1st ‘𝑔) = (1st ‘𝐺)) | |
3 | fveq2 6878 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
4 | 1, 2, 3 | ifbieq12d 4550 | . . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
5 | df-vtx 28123 | . . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) | |
6 | fvex 6891 | . . . 4 ⊢ (1st ‘𝐺) ∈ V | |
7 | fvex 6891 | . . . 4 ⊢ (Base‘𝐺) ∈ V | |
8 | 6, 7 | ifex 4572 | . . 3 ⊢ if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) ∈ V |
9 | 4, 5, 8 | fvmpt 6984 | . 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
10 | fvprc 6870 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
11 | prcnel 3496 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
12 | 11 | iffalsed 4533 | . . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺)) |
13 | fvprc 6870 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
14 | 10, 12, 13 | 3eqtr4rd 2782 | . 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
15 | 9, 14 | pm2.61i 182 | 1 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∅c0 4318 ifcif 4522 × cxp 5667 ‘cfv 6532 1st c1st 7955 Basecbs 17126 Vtxcvtx 28121 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fv 6540 df-vtx 28123 |
This theorem is referenced by: opvtxval 28128 funvtxdmge2val 28136 funvtxdm2val 28138 snstrvtxval 28162 vtxval0 28164 |
Copyright terms: Public domain | W3C validator |