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Mirrors > Home > MPE Home > Th. List > structgrssvtx | Structured version Visualization version GIF version |
Description: The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
structgrssvtx.g | ⊢ (𝜑 → 𝐺 Struct 𝑋) |
structgrssvtx.v | ⊢ (𝜑 → 𝑉 ∈ 𝑌) |
structgrssvtx.e | ⊢ (𝜑 → 𝐸 ∈ 𝑍) |
structgrssvtx.s | ⊢ (𝜑 → {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺) |
Ref | Expression |
---|---|
structgrssvtx | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | structgrssvtx.g | . 2 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
2 | structgrssvtx.v | . . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑌) | |
3 | structgrssvtx.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑍) | |
4 | structgrssvtx.s | . . 3 ⊢ (𝜑 → {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺) | |
5 | 1, 2, 3, 4 | structgrssvtxlem 26971 | . 2 ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺)) |
6 | opex 5323 | . . . . 5 ⊢ 〈(Base‘ndx), 𝑉〉 ∈ V | |
7 | opex 5323 | . . . . 5 ⊢ 〈(.ef‘ndx), 𝐸〉 ∈ V | |
8 | 6, 7 | prss 4709 | . . . 4 ⊢ ((〈(Base‘ndx), 𝑉〉 ∈ 𝐺 ∧ 〈(.ef‘ndx), 𝐸〉 ∈ 𝐺) ↔ {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺) |
9 | simpl 486 | . . . 4 ⊢ ((〈(Base‘ndx), 𝑉〉 ∈ 𝐺 ∧ 〈(.ef‘ndx), 𝐸〉 ∈ 𝐺) → 〈(Base‘ndx), 𝑉〉 ∈ 𝐺) | |
10 | 8, 9 | sylbir 238 | . . 3 ⊢ ({〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺 → 〈(Base‘ndx), 𝑉〉 ∈ 𝐺) |
11 | 4, 10 | syl 17 | . 2 ⊢ (𝜑 → 〈(Base‘ndx), 𝑉〉 ∈ 𝐺) |
12 | 1, 5, 2, 11 | basvtxval 26964 | 1 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3844 {cpr 4519 〈cop 4523 class class class wbr 5031 ‘cfv 6340 Struct cstr 16585 ndxcnx 16586 Basecbs 16589 .efcedgf 26937 Vtxcvtx 26944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-oadd 8138 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-dju 9406 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-xnn0 12052 df-z 12066 df-dec 12183 df-uz 12328 df-fz 12985 df-hash 13786 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-edgf 26938 df-vtx 26946 |
This theorem is referenced by: (None) |
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