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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 27075 | . . . 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 1928 | . 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph)) |
11 | 1 | fvexi 6684 | . . 3 ⊢ 𝑉 ∈ V |
12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V) |
13 | 2 | fvexi 6684 | . . . 4 ⊢ 𝐸 ∈ V |
14 | 13 | resex 5899 | . . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V |
15 | 14 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V) |
16 | 10, 12, 15 | gropeld 26818 | 1 ⊢ (𝐺 ∈ UPGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 ↾ cres 5557 ‘cfv 6355 Vtxcvtx 26781 iEdgciedg 26782 UPGraphcupgr 26865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-1st 7689 df-2nd 7690 df-vtx 26783 df-iedg 26784 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-subgr 27050 |
This theorem is referenced by: (None) |
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