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Mirrors > Home > MPE Home > Th. List > usgrexi | Structured version Visualization version GIF version |
Description: An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 10-Nov-2021.) |
Ref | Expression |
---|---|
usgrexi.p | ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} |
Ref | Expression |
---|---|
usgrexi | ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgrexi.p | . . . 4 ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | |
2 | 1 | usgrexilem 26893 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
3 | 1 | cusgrexilem1 26892 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V) |
4 | opiedgfv 26463 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = ( I ↾ 𝑃)) | |
5 | 3, 4 | mpdan 683 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = ( I ↾ 𝑃)) |
6 | 5 | dmeqd 5652 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = dom ( I ↾ 𝑃)) |
7 | opvtxfv 26460 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝑉) | |
8 | 3, 7 | mpdan 683 | . . . . . 6 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝑉) |
9 | 8 | pweqd 4452 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝒫 𝑉) |
10 | 9 | rabeqdv 3424 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
11 | 5, 6, 10 | f1eq123d 6468 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ((iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2} ↔ ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) |
12 | 2, 11 | mpbird 258 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2}) |
13 | opex 5241 | . . 3 ⊢ 〈𝑉, ( I ↾ 𝑃)〉 ∈ V | |
14 | eqid 2793 | . . . 4 ⊢ (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) | |
15 | eqid 2793 | . . . 4 ⊢ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) | |
16 | 14, 15 | isusgrs 26612 | . . 3 ⊢ (〈𝑉, ( I ↾ 𝑃)〉 ∈ V → (〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph ↔ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2})) |
17 | 13, 16 | mp1i 13 | . 2 ⊢ (𝑉 ∈ 𝑊 → (〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph ↔ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2})) |
18 | 12, 17 | mpbird 258 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1520 ∈ wcel 2079 {crab 3107 Vcvv 3432 𝒫 cpw 4447 〈cop 4472 I cid 5339 dom cdm 5435 ↾ cres 5437 –1-1→wf1 6214 ‘cfv 6217 2c2 11529 ♯chash 13528 Vtxcvtx 26452 iEdgciedg 26453 USGraphcusgr 26605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-n0 11735 df-z 11819 df-uz 12083 df-fz 12732 df-hash 13529 df-vtx 26454 df-iedg 26455 df-usgr 26607 |
This theorem is referenced by: cusgrexi 26896 |
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