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Mirrors > Home > MPE Home > Th. List > opvtxval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
opvtxval | ⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxval 26787 | . 2 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) | |
2 | iftrue 4475 | . 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (1st ‘𝐺)) | |
3 | 1, 2 | syl5eq 2870 | 1 ⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 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: opvtxfv 26791 |
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