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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 26617 | . 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 26618 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
8 | 7 | biimpd 221 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
9 | 8 | ancoms 452 | . 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 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 𝒫 cpw 4379 class class class wbr 4886 dom cdm 5355 ↾ cres 5357 ‘cfv 6135 Vtxcvtx 26344 iEdgciedg 26345 Edgcedg 26395 SubGraph csubgr 26614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-dm 5365 df-res 5367 df-iota 6099 df-fv 6143 df-subgr 26615 |
This theorem is referenced by: subgrprop2 26621 |
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