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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 6106 | . . 3 ⊢ (Vtx‘𝐻) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
3 | fvex 6113 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
4 | fvex 6113 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
5 | opvtxfv 25681 | . . . 4 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺)) | |
6 | 3, 4, 5 | mp2an 704 | . . 3 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
7 | 2, 6 | eqtr2i 2633 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻) |
8 | 1 | fveq2i 6106 | . . 3 ⊢ (iEdg‘𝐻) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
9 | opiedgfv 25684 | . . . 4 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺)) | |
10 | 3, 4, 9 | mp2an 704 | . . 3 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
11 | 8, 10 | eqtr2i 2633 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘𝐻) |
12 | 7, 11 | pm3.2i 470 | 1 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-2nd 7060 df-vtx 25675 df-iedg 25676 |
This theorem is referenced by: (None) |
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