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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 26790 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (Vtx‘𝑆) = (Base‘𝑆)) | |
6 | 3, 4, 5 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆)) |
7 | grstructd.b | . . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉) | |
8 | 6, 7 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
9 | funiedgdmge2val 26791 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (iEdg‘𝑆) = (.ef‘𝑆)) | |
10 | 3, 4, 9 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆)) |
11 | grstructd.e | . . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) | |
12 | 10, 11 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸) |
13 | 8, 12 | jca 514 | . 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)) |
14 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑔𝑆 | |
15 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) | |
16 | nfsbc1v 3791 | . . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓 | |
17 | 15, 16 | nfim 1893 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓) |
18 | fveqeq2 6673 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉)) | |
19 | fveqeq2 6673 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘𝑆) = 𝐸)) | |
20 | 18, 19 | anbi12d 632 | . . . 4 ⊢ (𝑔 = 𝑆 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸))) |
21 | sbceq1a 3782 | . . . 4 ⊢ (𝑔 = 𝑆 → (𝜓 ↔ [𝑆 / 𝑔]𝜓)) | |
22 | 20, 21 | imbi12d 347 | . . 3 ⊢ (𝑔 = 𝑆 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓))) |
23 | 14, 17, 22 | spcgf 3589 | . 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 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 [wsbc 3771 ∖ cdif 3932 ∅c0 4290 {csn 4560 class class class wbr 5058 dom cdm 5549 Fun wfun 6343 ‘cfv 6349 ≤ cle 10670 2c2 11686 ♯chash 13684 Basecbs 16477 .efcedgf 26768 Vtxcvtx 26775 iEdgciedg 26776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-vtx 26777 df-iedg 26778 |
This theorem is referenced by: grstructeld 26813 |
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