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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 28817 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (Vtx‘𝑆) = (Base‘𝑆)) | |
6 | 3, 4, 5 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆)) |
7 | grstructd.b | . . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉) | |
8 | 6, 7 | eqtrd 2768 | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
9 | funiedgdmge2val 28818 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (iEdg‘𝑆) = (.ef‘𝑆)) | |
10 | 3, 4, 9 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆)) |
11 | grstructd.e | . . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) | |
12 | 10, 11 | eqtrd 2768 | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸) |
13 | 8, 12 | jca 511 | . 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)) |
14 | nfcv 2899 | . . 3 ⊢ Ⅎ𝑔𝑆 | |
15 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) | |
16 | nfsbc1v 3795 | . . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓 | |
17 | 15, 16 | nfim 1892 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓) |
18 | fveqeq2 6900 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉)) | |
19 | fveqeq2 6900 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘𝑆) = 𝐸)) | |
20 | 18, 19 | anbi12d 631 | . . . 4 ⊢ (𝑔 = 𝑆 → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸))) |
21 | sbceq1a 3786 | . . . 4 ⊢ (𝑔 = 𝑆 → (𝜓 ↔ [𝑆 / 𝑔]𝜓)) | |
22 | 20, 21 | imbi12d 344 | . . 3 ⊢ (𝑔 = 𝑆 → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓))) |
23 | 14, 17, 22 | spcgf 3577 | . 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 395 ∀wal 1532 = wceq 1534 ∈ wcel 2099 [wsbc 3775 ∖ cdif 3942 ∅c0 4318 {csn 4624 class class class wbr 5142 dom cdm 5672 Fun wfun 6536 ‘cfv 6542 ≤ cle 11273 2c2 12291 ♯chash 14315 Basecbs 17173 .efcedgf 28792 Vtxcvtx 28802 iEdgciedg 28803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-n0 12497 df-xnn0 12569 df-z 12583 df-uz 12847 df-fz 13511 df-hash 14316 df-vtx 28804 df-iedg 28805 |
This theorem is referenced by: grstructeld 28840 |
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