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Mirrors > Home > MPE Home > Th. List > usgrop | Structured version Visualization version GIF version |
Description: A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.) |
Ref | Expression |
---|---|
usgrop | ⊢ (𝐺 ∈ USGraph → 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2732 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | usgrfs 28346 | . 2 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
4 | fvex 6892 | . . . 4 ⊢ (Vtx‘𝐺) ∈ V | |
5 | fvex 6892 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
6 | 4, 5 | pm3.2i 471 | . . 3 ⊢ ((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) |
7 | isusgrop 28351 | . . 3 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})) | |
8 | 6, 7 | mp1i 13 | . 2 ⊢ (𝐺 ∈ USGraph → (〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})) |
9 | 3, 8 | mpbird 256 | 1 ⊢ (𝐺 ∈ USGraph → 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 𝒫 cpw 4597 〈cop 4629 dom cdm 5670 –1-1→wf1 6530 ‘cfv 6533 2c2 12251 ♯chash 14274 Vtxcvtx 28185 iEdgciedg 28186 USGraphcusgr 28338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-hash 14275 df-vtx 28187 df-iedg 28188 df-usgr 28340 |
This theorem is referenced by: fusgrfis 28516 cusgrop 28624 |
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