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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 29457 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
| 3 | 1 | cusgrexilem1 29456 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃) ∈ V) |
| 4 | opiedgfv 29024 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = ( I ↾ 𝑃)) | |
| 5 | 3, 4 | mpdan 687 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = ( I ↾ 𝑃)) |
| 6 | 5 | dmeqd 5916 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = dom ( I ↾ 𝑃)) |
| 7 | opvtxfv 29021 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑃) ∈ V) → (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝑉) | |
| 8 | 3, 7 | mpdan 687 | . . . . . 6 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝑉) |
| 9 | 8 | pweqd 4617 | . . . . 5 ⊢ (𝑉 ∈ 𝑊 → 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = 𝒫 𝑉) |
| 10 | 9 | rabeqdv 3452 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) |
| 11 | 5, 6, 10 | f1eq123d 6840 | . . 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 257 | . 2 ⊢ (𝑉 ∈ 𝑊 → (iEdg‘〈𝑉, ( I ↾ 𝑃)〉):dom (iEdg‘〈𝑉, ( I ↾ 𝑃)〉)–1-1→{𝑥 ∈ 𝒫 (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) ∣ (♯‘𝑥) = 2}) |
| 13 | opex 5469 | . . 3 ⊢ 〈𝑉, ( I ↾ 𝑃)〉 ∈ V | |
| 14 | eqid 2737 | . . . 4 ⊢ (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) = (Vtx‘〈𝑉, ( I ↾ 𝑃)〉) | |
| 15 | eqid 2737 | . . . 4 ⊢ (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) = (iEdg‘〈𝑉, ( I ↾ 𝑃)〉) | |
| 16 | 14, 15 | isusgrs 29173 | . . 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 257 | 1 ⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ( I ↾ 𝑃)〉 ∈ USGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 𝒫 cpw 4600 〈cop 4632 I cid 5577 dom cdm 5685 ↾ cres 5687 –1-1→wf1 6558 ‘cfv 6561 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 USGraphcusgr 29166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-vtx 29015 df-iedg 29016 df-usgr 29168 |
| This theorem is referenced by: cusgrexi 29460 |
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