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Mirrors > Home > MPE Home > Th. List > upgrspan | Structured version Visualization version GIF version |
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 28865 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
9 | 2, 3, 4, 5, 6, 8 | uhgrspansubgr 29051 | . 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
10 | subupgr 29047 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph) | |
11 | 1, 9, 10 | syl2anc 583 | 1 ⊢ (𝜑 → 𝑆 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ↾ cres 5671 ‘cfv 6536 Vtxcvtx 28759 iEdgciedg 28760 UHGraphcuhgr 28819 UPGraphcupgr 28843 SubGraph csubgr 29027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-edg 28811 df-uhgr 28821 df-upgr 28845 df-subgr 29028 |
This theorem is referenced by: upgrspanop 29057 |
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