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Mirrors > Home > MPE Home > Th. List > grstructd | Structured version Visualization version GIF version |
Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.) |
Ref | Expression |
---|---|
gropd.g | ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓)) |
gropd.v | ⊢ (𝜑 → 𝑉 ∈ 𝑈) |
gropd.e | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
grstructd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑋) |
grstructd.f | ⊢ (𝜑 → Fun (𝑆 ∖ {∅})) |
grstructd.d | ⊢ (𝜑 → 2 ≤ (♯‘dom 𝑆)) |
grstructd.b | ⊢ (𝜑 → (Base‘𝑆) = 𝑉) |
grstructd.e | ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) |
Ref | Expression |
---|---|
grstructd | ⊢ (𝜑 → [𝑆 / 𝑔]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grstructd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋) | |
2 | gropd.g | . 2 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓)) | |
3 | grstructd.f | . . . . 5 ⊢ (𝜑 → Fun (𝑆 ∖ {∅})) | |
4 | grstructd.d | . . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘dom 𝑆)) | |
5 | funvtxdmge2val 26804 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (Vtx‘𝑆) = (Base‘𝑆)) | |
6 | 3, 4, 5 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆)) |
7 | grstructd.b | . . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉) | |
8 | 6, 7 | eqtrd 2833 | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
9 | funiedgdmge2val 26805 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (iEdg‘𝑆) = (.ef‘𝑆)) | |
10 | 3, 4, 9 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆)) |
11 | grstructd.e | . . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) | |
12 | 10, 11 | eqtrd 2833 | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸) |
13 | 8, 12 | jca 515 | . 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)) |
14 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑔𝑆 | |
15 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) | |
16 | nfsbc1v 3740 | . . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓 | |
17 | 15, 16 | nfim 1897 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓) |
18 | fveqeq2 6654 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉)) | |
19 | fveqeq2 6654 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘𝑆) = 𝐸)) | |
20 | 18, 19 | anbi12d 633 | . . . 4 ⊢ (𝑔 = 𝑆 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸))) |
21 | sbceq1a 3731 | . . . 4 ⊢ (𝑔 = 𝑆 → (𝜓 ↔ [𝑆 / 𝑔]𝜓)) | |
22 | 20, 21 | imbi12d 348 | . . 3 ⊢ (𝑔 = 𝑆 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓))) |
23 | 14, 17, 22 | spcgf 3538 | . 2 ⊢ (𝑆 ∈ 𝑋 → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓))) |
24 | 1, 2, 13, 23 | syl3c 66 | 1 ⊢ (𝜑 → [𝑆 / 𝑔]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 [wsbc 3720 ∖ cdif 3878 ∅c0 4243 {csn 4525 class class class wbr 5030 dom cdm 5519 Fun wfun 6318 ‘cfv 6324 ≤ cle 10665 2c2 11680 ♯chash 13686 Basecbs 16475 .efcedgf 26782 Vtxcvtx 26789 iEdgciedg 26790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-vtx 26791 df-iedg 26792 |
This theorem is referenced by: grstructeld 26827 |
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