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Mirrors > Home > MPE Home > Th. List > opiedgval | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.) |
Ref | Expression |
---|---|
opiedgval | ⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iedgval 28250 | . 2 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) | |
2 | iftrue 4533 | . 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (2nd ‘𝐺)) | |
3 | 1, 2 | eqtrid 2784 | 1 ⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ifcif 4527 × cxp 5673 ‘cfv 6540 2nd c2nd 7970 .efcedgf 28235 iEdgciedg 28246 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-iedg 28248 |
This theorem is referenced by: opiedgfv 28256 |
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