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Theorem subgrprop2 27064

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.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
subgrprop2	⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))

**Proof of Theorem subgrprop2**
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 27063	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		resss 5843	. . . 4 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8		sseq1 3940	. . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
9	7, 8	mpbiri 261	. . 3 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵)
10	9	3anim2i 1150	. 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 1084 = wceq 1538 ⊆ wss 3881 𝒫 cpw 4497 class class class wbr 5030 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 SubGraph csubgr 27057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-res 5531 df-iota 6283 df-fv 6332 df-subgr 27058

This theorem is referenced by: uhgrissubgr 27065 subgrprop3 27066 subgrfun 27071 subgreldmiedg 27073 subgruhgredgd 27074 subumgredg2 27075 subuhgr 27076 subupgr 27077 subumgr 27078 subusgr 27079 subgrwlk 32492

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