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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 29032 | . 2 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) | |
2 | iftrue 4537 | . 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st ‘𝐺), (Base‘𝐺)) = (1st ‘𝐺)) | |
3 | 1, 2 | eqtrid 2787 | 1 ⊢ (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ifcif 4531 × cxp 5687 ‘cfv 6563 1st c1st 8011 Basecbs 17245 Vtxcvtx 29028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-vtx 29030 |
This theorem is referenced by: opvtxfv 29036 |
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