![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 28935 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
9 | 2, 3, 4, 5, 6, 8 | uhgrspansubgr 29124 | . 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
10 | subupgr 29120 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph) | |
11 | 1, 9, 10 | syl2anc 582 | 1 ⊢ (𝜑 → 𝑆 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ↾ cres 5684 ‘cfv 6553 Vtxcvtx 28829 iEdgciedg 28830 UHGraphcuhgr 28889 UPGraphcupgr 28913 SubGraph csubgr 29100 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-edg 28881 df-uhgr 28891 df-upgr 28915 df-subgr 29101 |
This theorem is referenced by: upgrspanop 29130 |
Copyright terms: Public domain | W3C validator |