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Mirrors > Home > MPE Home > Th. List > usgrspanop | Structured version Visualization version GIF version |
Description: A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
uhgrspanop.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrspanop.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
usgrspanop | ⊢ (𝐺 ∈ USGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ USGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspanop.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgrspanop.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | vex 3441 | . . . . . 6 ⊢ 𝑔 ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ V) |
5 | simprl 769 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (Vtx‘𝑔) = 𝑉) | |
6 | simprr 771 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) | |
7 | simpl 484 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝐺 ∈ USGraph) | |
8 | 1, 2, 4, 5, 6, 7 | usgrspan 27707 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ USGraph) |
9 | 8 | ex 414 | . . 3 ⊢ (𝐺 ∈ USGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ USGraph)) |
10 | 9 | alrimiv 1928 | . 2 ⊢ (𝐺 ∈ USGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ USGraph)) |
11 | 1 | fvexi 6818 | . . 3 ⊢ 𝑉 ∈ V |
12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ USGraph → 𝑉 ∈ V) |
13 | 2 | fvexi 6818 | . . . 4 ⊢ 𝐸 ∈ V |
14 | 13 | resex 5951 | . . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V |
15 | 14 | a1i 11 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐸 ↾ 𝐴) ∈ V) |
16 | 10, 12, 15 | gropeld 27448 | 1 ⊢ (𝐺 ∈ USGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ⟨cop 4571 ↾ cres 5602 ‘cfv 6458 Vtxcvtx 27411 iEdgciedg 27412 USGraphcusgr 27564 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-hash 14091 df-vtx 27413 df-iedg 27414 df-edg 27463 df-uhgr 27473 df-upgr 27497 df-umgr 27498 df-uspgr 27565 df-usgr 27566 df-subgr 27680 |
This theorem is referenced by: (None) |
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