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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 29134 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
9 | 2, 3, 4, 5, 6, 8 | uhgrspansubgr 29323 | . 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
10 | subupgr 29319 | . 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 1537 ∈ wcel 2106 class class class wbr 5148 ↾ cres 5691 ‘cfv 6563 Vtxcvtx 29028 iEdgciedg 29029 UHGraphcuhgr 29088 UPGraphcupgr 29112 SubGraph csubgr 29299 |
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-edg 29080 df-uhgr 29090 df-upgr 29114 df-subgr 29300 |
This theorem is referenced by: upgrspanop 29329 |
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