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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 40497 | . 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) |
7 | resss 5342 | . . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵 | |
8 | sseq1 3589 | . . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵)) | |
9 | 7, 8 | mpbiri 247 | . . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵) |
10 | 9 | 3anim2i 1242 | . 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 1031 = wceq 1475 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 Edgcedga 25792 SubGraph csubgr 40491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-res 5050 df-iota 5768 df-fv 5812 df-subgr 40492 |
This theorem is referenced by: uhgrissubgr 40499 subgrprop3 40500 subgrfun 40505 subgreldmiedg 40507 subgruhgredgd 40508 subumgredg2 40509 subuhgr 40510 subupgr 40511 subumgr 40512 subusgr 40513 |
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