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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 26822 | . . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
7 | usgrstrrepe.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺) | |
8 | 6, 7 | syl6eqr 2876 | . . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝑉) |
9 | 8 | pweqd 4560 | . . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝒫 𝑉) |
10 | 9 | rabeqdv 3486 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
11 | f1eq3 6574 | . . . . 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 259 | . . 3 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2}) |
14 | 2, 3, 4, 5 | setsiedg 26823 | . . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) |
15 | 14 | dmeqd 5776 | . . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = dom 𝐸) |
16 | eqidd 2824 | . . . 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 6610 | . . 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 259 | . 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 7191 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
20 | eqid 2823 | . . . 4 ⊢ (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
21 | eqid 2823 | . . . 4 ⊢ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
22 | 20, 21 | isusgrs 26943 | . . 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 259 | 1 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 𝒫 cpw 4541 〈cop 4575 class class class wbr 5068 dom cdm 5557 –1-1→wf1 6354 ‘cfv 6357 (class class class)co 7158 2c2 11695 ♯chash 13693 Struct cstr 16481 ndxcnx 16482 sSet csts 16483 Basecbs 16485 .efcedgf 26776 Vtxcvtx 26783 iEdgciedg 26784 USGraphcusgr 26936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-edgf 26777 df-vtx 26785 df-iedg 26786 df-usgr 26938 |
This theorem is referenced by: structtousgr 27229 |
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