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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 6895 | . . 3 ⊢ (Vtx‘𝐻) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) |
3 | fvex 6905 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
4 | fvex 6905 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
5 | 3, 4 | opvtxfvi 28269 | . . 3 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺) |
6 | 2, 5 | eqtr2i 2762 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻) |
7 | 1 | fveq2i 6895 | . . 3 ⊢ (iEdg‘𝐻) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) |
8 | 3, 4 | opiedgfvi 28270 | . . 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 4635 ‘cfv 6544 Vtxcvtx 28256 iEdgciedg 28257 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fv 6552 df-1st 7975 df-2nd 7976 df-vtx 28258 df-iedg 28259 |
This theorem is referenced by: (None) |
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