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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 29066 | . . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
7 | usgrstrrepe.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺) | |
8 | 6, 7 | eqtr4di 2792 | . . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝑉) |
9 | 8 | pweqd 4621 | . . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝒫 𝑉) |
10 | 9 | rabeqdv 3448 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
11 | f1eq3 6801 | . . . . 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 257 | . . 3 ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∣ (♯‘𝑥) = 2}) |
14 | 2, 3, 4, 5 | setsiedg 29067 | . . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) |
15 | 14 | dmeqd 5918 | . . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = dom 𝐸) |
16 | eqidd 2735 | . . . 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 6840 | . . 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 257 | . 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 7463 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
20 | eqid 2734 | . . . 4 ⊢ (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
21 | eqid 2734 | . . . 4 ⊢ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
22 | 20, 21 | isusgrs 29187 | . . 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 257 | 1 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 𝒫 cpw 4604 〈cop 4636 class class class wbr 5147 dom cdm 5688 –1-1→wf1 6559 ‘cfv 6562 (class class class)co 7430 2c2 12318 ♯chash 14365 Struct cstr 17179 sSet csts 17196 ndxcnx 17226 Basecbs 17244 .efcedgf 29017 Vtxcvtx 29027 iEdgciedg 29028 USGraphcusgr 29180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-edgf 29018 df-vtx 29029 df-iedg 29030 df-usgr 29182 |
This theorem is referenced by: structtousgr 29476 |
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