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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 3482 | . . . . . 6 ⊢ 𝑔 ∈ V | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ V) |
5 | simprl 771 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (Vtx‘𝑔) = 𝑉) | |
6 | simprr 773 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) | |
7 | simpl 482 | . . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝐺 ∈ UPGraph) | |
8 | 1, 2, 4, 5, 6, 7 | upgrspan 29325 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ UPGraph) |
9 | 8 | ex 412 | . . 3 ⊢ (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph)) |
10 | 9 | alrimiv 1925 | . 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph)) |
11 | 1 | fvexi 6921 | . . 3 ⊢ 𝑉 ∈ V |
12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V) |
13 | 2 | fvexi 6921 | . . . 4 ⊢ 𝐸 ∈ V |
14 | 13 | resex 6049 | . . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V |
15 | 14 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V) |
16 | 10, 12, 15 | gropeld 29065 | 1 ⊢ (𝐺 ∈ UPGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↾ cres 5691 ‘cfv 6563 Vtxcvtx 29028 iEdgciedg 29029 UPGraphcupgr 29112 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1st 8013 df-2nd 8014 df-vtx 29030 df-iedg 29031 df-edg 29080 df-uhgr 29090 df-upgr 29114 df-subgr 29300 |
This theorem is referenced by: (None) |
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