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| Description: A spanning subgraph 𝑆 of a pseudograph 𝐺 is a pseudograph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.) | 
| Ref | Expression | 
|---|---|
| uhgrspan.v | ⊢ 𝑉 = (Vtx‘𝐺) | 
| uhgrspan.e | ⊢ 𝐸 = (iEdg‘𝐺) | 
| uhgrspan.s | ⊢ (𝜑 → 𝑆 ∈ 𝑊) | 
| uhgrspan.q | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | 
| uhgrspan.r | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | 
| upgrspan.g | ⊢ (𝜑 → 𝐺 ∈ UPGraph) | 
| Ref | Expression | 
|---|---|
| upgrspan | ⊢ (𝜑 → 𝑆 ∈ UPGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | upgrspan.g | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | |
| 2 | uhgrspan.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | uhgrspan.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 4 | uhgrspan.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
| 5 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
| 6 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
| 7 | upgruhgr 29120 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
| 8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | 
| 9 | 2, 3, 4, 5, 6, 8 | uhgrspansubgr 29309 | . 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) | 
| 10 | subupgr 29305 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph) | |
| 11 | 1, 9, 10 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑆 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ↾ cres 5686 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 UHGraphcuhgr 29074 UPGraphcupgr 29098 SubGraph csubgr 29285 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-edg 29066 df-uhgr 29076 df-upgr 29100 df-subgr 29286 | 
| This theorem is referenced by: upgrspanop 29315 | 
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