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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 3484 | . . . . . 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 29310 | . . . 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 1927 | . 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph)) | 
| 11 | 1 | fvexi 6920 | . . 3 ⊢ 𝑉 ∈ V | 
| 12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V) | 
| 13 | 2 | fvexi 6920 | . . . 4 ⊢ 𝐸 ∈ V | 
| 14 | 13 | resex 6047 | . . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V | 
| 15 | 14 | a1i 11 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V) | 
| 16 | 10, 12, 15 | gropeld 29050 | 1 ⊢ (𝐺 ∈ UPGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↾ cres 5687 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UPGraphcupgr 29097 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-1st 8014 df-2nd 8015 df-vtx 29015 df-iedg 29016 df-edg 29065 df-uhgr 29075 df-upgr 29099 df-subgr 29285 | 
| This theorem is referenced by: (None) | 
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