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Mirrors > Home > MPE Home > Th. List > graop | Structured version Visualization version GIF version |
Description: Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.) |
Ref | Expression |
---|---|
graop.h | ⊢ 𝐻 = 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 |
Ref | Expression |
---|---|
graop | ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | graop.h | . . . 4 ⊢ 𝐻 = 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 | |
2 | 1 | fveq2i 6891 | . . 3 ⊢ (Vtx‘𝐻) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
3 | fvex 6901 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
4 | fvex 6901 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
5 | 3, 4 | opvtxfvi 28249 | . . 3 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
6 | 2, 5 | eqtr2i 2762 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻) |
7 | 1 | fveq2i 6891 | . . 3 ⊢ (iEdg‘𝐻) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | 3, 4 | opiedgfvi 28250 | . . 3 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
9 | 7, 8 | eqtr2i 2762 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘𝐻) |
10 | 6, 9 | pm3.2i 472 | 1 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 〈cop 4633 ‘cfv 6540 Vtxcvtx 28236 iEdgciedg 28237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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-rn 5686 df-iota 6492 df-fun 6542 df-fv 6548 df-1st 7970 df-2nd 7971 df-vtx 28238 df-iedg 28239 |
This theorem is referenced by: (None) |
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