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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 6923 | . . 3 ⊢ (Vtx‘𝐻) = (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
3 | fvex 6933 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
4 | fvex 6933 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
5 | 3, 4 | opvtxfvi 29044 | . . 3 ⊢ (Vtx‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (Vtx‘𝐺) |
6 | 2, 5 | eqtr2i 2769 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘𝐻) |
7 | 1 | fveq2i 6923 | . . 3 ⊢ (iEdg‘𝐻) = (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) |
8 | 3, 4 | opiedgfvi 29045 | . . 3 ⊢ (iEdg‘〈(Vtx‘𝐺), (iEdg‘𝐺)〉) = (iEdg‘𝐺) |
9 | 7, 8 | eqtr2i 2769 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘𝐻) |
10 | 6, 9 | pm3.2i 470 | 1 ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 〈cop 4654 ‘cfv 6573 Vtxcvtx 29031 iEdgciedg 29032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-1st 8030 df-2nd 8031 df-vtx 29033 df-iedg 29034 |
This theorem is referenced by: (None) |
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