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| Mirrors > Home > MPE Home > Th. List > opvtxfvi | Structured version Visualization version GIF version | ||
| Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.) |
| Ref | Expression |
|---|---|
| opvtxfvi.v | ⊢ 𝑉 ∈ V |
| opvtxfvi.e | ⊢ 𝐸 ∈ V |
| Ref | Expression |
|---|---|
| opvtxfvi | ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opvtxfvi.v | . 2 ⊢ 𝑉 ∈ V | |
| 2 | opvtxfvi.e | . 2 ⊢ 𝐸 ∈ V | |
| 3 | opvtxfv 29021 | . 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 Vtxcvtx 29013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-vtx 29015 |
| This theorem is referenced by: graop 29046 vtxvalsnop 29058 uhgrspanop 29313 fusgrfis 29347 cusgrsize 29472 fusgrmaxsize 29482 vtxdgop 29488 vtxdginducedm1 29561 vtxdginducedm1fi 29562 finsumvtxdg2ssteplem4 29566 finsumvtxdg2size 29568 eupth2lem3 30255 konigsberglem1 30271 konigsberglem2 30272 konigsberglem3 30273 |
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