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| 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 29288 | . 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 29289 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) | 
| 8 | 7 | biimpd 229 | . . 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 1086 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4599 class class class wbr 5142 dom cdm 5684 ↾ cres 5686 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 Edgcedg 29065 SubGraph csubgr 29285 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dm 5694 df-res 5696 df-iota 6513 df-fv 6568 df-subgr 29286 | 
| This theorem is referenced by: subgrprop2 29292 subgrwlk 35138 | 
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