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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 27057 | . 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) |
7 | resss 5880 | . . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵 | |
8 | sseq1 3994 | . . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵)) | |
9 | 7, 8 | mpbiri 260 | . . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵) |
10 | 9 | 3anim2i 1149 | . 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 1083 = wceq 1537 ⊆ wss 3938 𝒫 cpw 4541 class class class wbr 5068 dom cdm 5557 ↾ cres 5559 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 Edgcedg 26834 SubGraph csubgr 27051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-dm 5567 df-res 5569 df-iota 6316 df-fv 6365 df-subgr 27052 |
This theorem is referenced by: uhgrissubgr 27059 subgrprop3 27060 subgrfun 27065 subgreldmiedg 27067 subgruhgredgd 27068 subumgredg2 27069 subuhgr 27070 subupgr 27071 subumgr 27072 subusgr 27073 subgrwlk 32381 |
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