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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 2819 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V))) | |
2 | fveq2 6892 | . . . 4 ⊢ (𝑔 = 𝐺 → (1st ‘𝑔) = (1st ‘𝐺)) | |
3 | fveq2 6892 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
4 | 1, 2, 3 | ifbieq12d 4557 | . . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
5 | df-vtx 28523 | . . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) | |
6 | fvex 6905 | . . . 4 ⊢ (1st ‘𝐺) ∈ V | |
7 | fvex 6905 | . . . 4 ⊢ (Base‘𝐺) ∈ V | |
8 | 6, 7 | ifex 4579 | . . 3 ⊢ if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) ∈ V |
9 | 4, 5, 8 | fvmpt 6999 | . 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
10 | fvprc 6884 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
11 | prcnel 3496 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
12 | 11 | iffalsed 4540 | . . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺)) |
13 | fvprc 6884 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
14 | 10, 12, 13 | 3eqtr4rd 2781 | . 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 1539 ∈ wcel 2104 Vcvv 3472 ∅c0 4323 ifcif 4529 × cxp 5675 ‘cfv 6544 1st c1st 7977 Basecbs 17150 Vtxcvtx 28521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-vtx 28523 |
This theorem is referenced by: opvtxval 28528 funvtxdmge2val 28536 funvtxdm2val 28538 snstrvtxval 28562 vtxval0 28564 |
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