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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 29127 | . 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 29128 | . . . 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 457 | . 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3939 𝒫 cpw 4598 class class class wbr 5143 dom cdm 5672 ↾ cres 5674 ‘cfv 6543 Vtxcvtx 28853 iEdgciedg 28854 Edgcedg 28904 SubGraph csubgr 29124 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
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-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-dm 5682 df-res 5684 df-iota 6495 df-fv 6551 df-subgr 29125 |
This theorem is referenced by: subgrprop2 29131 subgrwlk 34799 |
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