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Mirrors > Home > MPE Home > Th. List > subgrprop | Structured version Visualization version GIF version |
Description: The properties of a subgraph. (Contributed by AV, 19-Nov-2020.) |
Ref | Expression |
---|---|
issubgr.v | ⊢ 𝑉 = (Vtx‘𝑆) |
issubgr.a | ⊢ 𝐴 = (Vtx‘𝐺) |
issubgr.i | ⊢ 𝐼 = (iEdg‘𝑆) |
issubgr.b | ⊢ 𝐵 = (iEdg‘𝐺) |
issubgr.e | ⊢ 𝐸 = (Edg‘𝑆) |
Ref | Expression |
---|---|
subgrprop | ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrv 29035 | . 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V)) | |
2 | issubgr.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝑆) | |
3 | issubgr.a | . . . . 5 ⊢ 𝐴 = (Vtx‘𝐺) | |
4 | issubgr.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝑆) | |
5 | issubgr.b | . . . . 5 ⊢ 𝐵 = (iEdg‘𝐺) | |
6 | issubgr.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝑆) | |
7 | 2, 3, 4, 5, 6 | issubgr 29036 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
8 | 7 | biimpd 228 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
9 | 8 | ancoms 458 | . 2 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
10 | 1, 9 | mpcom 38 | 1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 class class class wbr 5141 dom cdm 5669 ↾ cres 5671 ‘cfv 6537 Vtxcvtx 28764 iEdgciedg 28765 Edgcedg 28815 SubGraph csubgr 29032 |
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-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 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-xp 5675 df-rel 5676 df-dm 5679 df-res 5681 df-iota 6489 df-fv 6545 df-subgr 29033 |
This theorem is referenced by: subgrprop2 29039 subgrwlk 34651 |
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