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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 27058 | . 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) |
7 | resss 5881 | . . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵 | |
8 | sseq1 3995 | . . . 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 1536 ⊆ wss 3939 𝒫 cpw 4542 class class class wbr 5069 dom cdm 5558 ↾ cres 5560 ‘cfv 6358 Vtxcvtx 26784 iEdgciedg 26785 Edgcedg 26835 SubGraph csubgr 27052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-dm 5568 df-res 5570 df-iota 6317 df-fv 6366 df-subgr 27053 |
This theorem is referenced by: uhgrissubgr 27060 subgrprop3 27061 subgrfun 27066 subgreldmiedg 27068 subgruhgredgd 27069 subumgredg2 27070 subuhgr 27071 subupgr 27072 subumgr 27073 subusgr 27074 subgrwlk 32383 |
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