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Mirrors > Home > MPE Home > Th. List > subgrprop2 | Structured version Visualization version GIF version |
Description: The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺, connecting vertices of the subgraph only. (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 |
---|---|
subgrprop2 | ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝑆) | |
2 | issubgr.a | . . 3 ⊢ 𝐴 = (Vtx‘𝐺) | |
3 | issubgr.i | . . 3 ⊢ 𝐼 = (iEdg‘𝑆) | |
4 | issubgr.b | . . 3 ⊢ 𝐵 = (iEdg‘𝐺) | |
5 | issubgr.e | . . 3 ⊢ 𝐸 = (Edg‘𝑆) | |
6 | 1, 2, 3, 4, 5 | subgrprop 29305 | . 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) |
7 | resss 6022 | . . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵 | |
8 | sseq1 4021 | . . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵)) | |
9 | 7, 8 | mpbiri 258 | . . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵) |
10 | 9 | 3anim2i 1152 | . 2 ⊢ ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)) |
11 | 6, 10 | syl 17 | 1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ⊆ wss 3963 𝒫 cpw 4605 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 SubGraph csubgr 29299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-res 5701 df-iota 6516 df-fv 6571 df-subgr 29300 |
This theorem is referenced by: uhgrissubgr 29307 subgrprop3 29308 subgrfun 29313 subgreldmiedg 29315 subgruhgredgd 29316 subumgredg2 29317 subuhgr 29318 subupgr 29319 subumgr 29320 subusgr 29321 subgrwlk 35117 |
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