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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 26747 | . . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
7 | usgrstrrepe.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺) | |
8 | 6, 7 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝑉) |
9 | 8 | pweqd 4540 | . . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝒫 𝑉) |
10 | 9 | rabeqdv 3482 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
11 | f1eq3 6565 | . . . . 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 258 | . . 3 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2}) |
14 | 2, 3, 4, 5 | setsiedg 26748 | . . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) |
15 | 14 | dmeqd 5767 | . . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = dom 𝐸) |
16 | eqidd 2819 | . . . 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 6601 | . . 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 258 | . 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 7178 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
20 | eqid 2818 | . . . 4 ⊢ (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
21 | eqid 2818 | . . . 4 ⊢ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
22 | 20, 21 | isusgrs 26868 | . . 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 258 | 1 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 𝒫 cpw 4535 〈cop 4563 class class class wbr 5057 dom cdm 5548 –1-1→wf1 6345 ‘cfv 6348 (class class class)co 7145 2c2 11680 ♯chash 13678 Struct cstr 16467 ndxcnx 16468 sSet csts 16469 Basecbs 16471 .efcedgf 26701 Vtxcvtx 26708 iEdgciedg 26709 USGraphcusgr 26861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-edgf 26702 df-vtx 26710 df-iedg 26711 df-usgr 26863 |
This theorem is referenced by: structtousgr 27154 |
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