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Mirrors > Home > MPE Home > Th. List > usgrstrrepe | Structured version Visualization version GIF version |
Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
usgrstrrepe.v | ⊢ 𝑉 = (Base‘𝐺) |
usgrstrrepe.i | ⊢ 𝐼 = (.ef‘ndx) |
usgrstrrepe.s | ⊢ (𝜑 → 𝐺 Struct 𝑋) |
usgrstrrepe.b | ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) |
usgrstrrepe.w | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
usgrstrrepe.e | ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
Ref | Expression |
---|---|
usgrstrrepe | ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrstrrepe.e | . . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | |
2 | usgrstrrepe.i | . . . . . . . . 9 ⊢ 𝐼 = (.ef‘ndx) | |
3 | usgrstrrepe.s | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
4 | usgrstrrepe.b | . . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) | |
5 | usgrstrrepe.w | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
6 | 2, 3, 4, 5 | setsvtx 28965 | . . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
7 | usgrstrrepe.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺) | |
8 | 6, 7 | eqtr4di 2784 | . . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝑉) |
9 | 8 | pweqd 4614 | . . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝒫 𝑉) |
10 | 9 | rabeqdv 3435 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
11 | f1eq3 6784 | . . . . 5 ⊢ ({𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} → (𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) |
13 | 1, 12 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2}) |
14 | 2, 3, 4, 5 | setsiedg 28966 | . . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) |
15 | 14 | dmeqd 5902 | . . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = dom 𝐸) |
16 | eqidd 2727 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2}) | |
17 | 14, 15, 16 | f1eq123d 6824 | . . 3 ⊢ (𝜑 → ((iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2})) |
18 | 13, 17 | mpbird 256 | . 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2}) |
19 | ovex 7446 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
20 | eqid 2726 | . . . 4 ⊢ (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
21 | eqid 2726 | . . . 4 ⊢ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
22 | 20, 21 | isusgrs 29086 | . . 3 ⊢ ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ V → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph ↔ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2})) |
23 | 19, 22 | mp1i 13 | . 2 ⊢ (𝜑 → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph ↔ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2})) |
24 | 18, 23 | mpbird 256 | 1 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 𝒫 cpw 4597 〈cop 4629 class class class wbr 5143 dom cdm 5672 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7413 2c2 12310 ♯chash 14339 Struct cstr 17140 sSet csts 17157 ndxcnx 17187 Basecbs 17205 .efcedgf 28916 Vtxcvtx 28926 iEdgciedg 28927 USGraphcusgr 29079 |
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 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9934 df-card 9972 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12516 df-xnn0 12588 df-z 12602 df-dec 12721 df-uz 12866 df-fz 13530 df-hash 14340 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-edgf 28917 df-vtx 28928 df-iedg 28929 df-usgr 29081 |
This theorem is referenced by: structtousgr 29375 |
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