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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 2902 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V))) | |
2 | fveq2 6672 | . . . 4 ⊢ (𝑔 = 𝐺 → (1st ‘𝑔) = (1st ‘𝐺)) | |
3 | fveq2 6672 | . . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺)) | |
4 | 1, 2, 3 | ifbieq12d 4496 | . . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
5 | df-vtx 26785 | . . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))) | |
6 | fvex 6685 | . . . 4 ⊢ (1st ‘𝐺) ∈ V | |
7 | fvex 6685 | . . . 4 ⊢ (Base‘𝐺) ∈ V | |
8 | 6, 7 | ifex 4517 | . . 3 ⊢ if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) ∈ V |
9 | 4, 5, 8 | fvmpt 6770 | . 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
10 | fvprc 6665 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅) | |
11 | prcnel 3520 | . . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V)) | |
12 | 11 | iffalsed 4480 | . . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺)) |
13 | fvprc 6665 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
14 | 10, 12, 13 | 3eqtr4rd 2869 | . 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺))) |
15 | 9, 14 | pm2.61i 184 | 1 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 ifcif 4469 × cxp 5555 ‘cfv 6357 1st c1st 7689 Basecbs 16485 Vtxcvtx 26783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-vtx 26785 |
This theorem is referenced by: opvtxval 26790 funvtxdmge2val 26798 funvtxdm2val 26800 snstrvtxval 26824 vtxval0 26826 |
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