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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 29043 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (Vtx‘𝑆) = (Base‘𝑆)) | |
6 | 3, 4, 5 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = (Base‘𝑆)) |
7 | grstructd.b | . . . 4 ⊢ (𝜑 → (Base‘𝑆) = 𝑉) | |
8 | 6, 7 | eqtrd 2775 | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
9 | funiedgdmge2val 29044 | . . . . 5 ⊢ ((Fun (𝑆 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝑆)) → (iEdg‘𝑆) = (.ef‘𝑆)) | |
10 | 3, 4, 9 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (iEdg‘𝑆) = (.ef‘𝑆)) |
11 | grstructd.e | . . . 4 ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) | |
12 | 10, 11 | eqtrd 2775 | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = 𝐸) |
13 | 8, 12 | jca 511 | . 2 ⊢ (𝜑 → ((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸)) |
14 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑔𝑆 | |
15 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑔((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) | |
16 | nfsbc1v 3811 | . . . 4 ⊢ Ⅎ𝑔[𝑆 / 𝑔]𝜓 | |
17 | 15, 16 | nfim 1894 | . . 3 ⊢ Ⅎ𝑔(((Vtx‘𝑆) = 𝑉 ∧ (iEdg‘𝑆) = 𝐸) → [𝑆 / 𝑔]𝜓) |
18 | fveqeq2 6916 | . . . . 5 ⊢ (𝑔 = 𝑆 → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘𝑆) = 𝑉)) | |
19 | fveqeq2 6916 | . . . . 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 3802 | . . . 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 3591 | . 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 1535 = wceq 1537 ∈ wcel 2106 [wsbc 3791 ∖ cdif 3960 ∅c0 4339 {csn 4631 class class class wbr 5148 dom cdm 5689 Fun wfun 6557 ‘cfv 6563 ≤ cle 11294 2c2 12319 ♯chash 14366 Basecbs 17245 .efcedgf 29018 Vtxcvtx 29028 iEdgciedg 29029 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-vtx 29030 df-iedg 29031 |
This theorem is referenced by: grstructeld 29066 |
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