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Mirrors > Home > MPE Home > Th. List > upgrspanop | Structured version Visualization version GIF version |
Description: A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
uhgrspanop.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrspanop.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgrspanop | ⊢ (𝐺 ∈ UPGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspanop.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgrspanop.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | vex 3497 | . . . . . 6 ⊢ 𝑔 ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ V) |
5 | simprl 769 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (Vtx‘𝑔) = 𝑉) | |
6 | simprr 771 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) | |
7 | simpl 485 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝐺 ∈ UPGraph) | |
8 | 1, 2, 4, 5, 6, 7 | upgrspan 27069 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ UPGraph) |
9 | 8 | ex 415 | . . 3 ⊢ (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph)) |
10 | 9 | alrimiv 1924 | . 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph)) |
11 | 1 | fvexi 6678 | . . 3 ⊢ 𝑉 ∈ V |
12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V) |
13 | 2 | fvexi 6678 | . . . 4 ⊢ 𝐸 ∈ V |
14 | 13 | resex 5893 | . . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V |
15 | 14 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V) |
16 | 10, 12, 15 | gropeld 26812 | 1 ⊢ (𝐺 ∈ UPGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4566 ↾ cres 5551 ‘cfv 6349 Vtxcvtx 26775 iEdgciedg 26776 UPGraphcupgr 26859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-1st 7683 df-2nd 7684 df-vtx 26777 df-iedg 26778 df-edg 26827 df-uhgr 26837 df-upgr 26861 df-subgr 27044 |
This theorem is referenced by: (None) |
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