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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 25768 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph ) | |
8 | 1, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph ) |
9 | 2, 3, 4, 5, 6, 8 | uhgrspansubgr 40515 | . 2 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
10 | subupgr 40511 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph ) | |
11 | 1, 9, 10 | syl2anc 691 | 1 ⊢ (𝜑 → 𝑆 ∈ UPGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↾ cres 5040 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 UPGraph cupgr 25747 SubGraph csubgr 40491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-subgr 40492 |
This theorem is referenced by: upgrspanop 40521 |
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