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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 26359 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (Vtx‘𝑆) = (Base‘𝑆)) | |
6 | 3, 4, 5 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆)) |
7 | grstructd.b | . . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉) | |
8 | 6, 7 | eqtrd 2814 | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
9 | funiedgdmge2val 26360 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (iEdg‘𝑆) = (.ef‘𝑆)) | |
10 | 3, 4, 9 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆)) |
11 | grstructd.e | . . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) | |
12 | 10, 11 | eqtrd 2814 | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸) |
13 | 8, 12 | jca 507 | . 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)) |
14 | nfcv 2934 | . . 3 ⊢ Ⅎ𝑔𝑆 | |
15 | nfv 1957 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) | |
16 | nfsbc1v 3672 | . . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓 | |
17 | 15, 16 | nfim 1943 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓) |
18 | fveqeq2 6455 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉)) | |
19 | fveqeq2 6455 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘𝑆) = 𝐸)) | |
20 | 18, 19 | anbi12d 624 | . . . 4 ⊢ (𝑔 = 𝑆 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸))) |
21 | sbceq1a 3663 | . . . 4 ⊢ (𝑔 = 𝑆 → (𝜓 ↔ [𝑆 / 𝑔]𝜓)) | |
22 | 20, 21 | imbi12d 336 | . . 3 ⊢ (𝑔 = 𝑆 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓))) |
23 | 14, 17, 22 | spcgf 3490 | . 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 386 ∀wal 1599 = wceq 1601 ∈ wcel 2107 [wsbc 3652 ∖ cdif 3789 ∅c0 4141 {csn 4398 class class class wbr 4886 dom cdm 5355 Fun wfun 6129 ‘cfv 6135 ≤ cle 10412 2c2 11430 ♯chash 13435 Basecbs 16255 .efcedgf 26337 Vtxcvtx 26344 iEdgciedg 26345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-hash 13436 df-vtx 26346 df-iedg 26347 |
This theorem is referenced by: grstructeld 26382 |
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