subgrprop - Intuitionistic Logic Explorer

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Theorem subgrprop 16380

Description: The properties of a subgraph. (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
subgrprop	⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

**Proof of Theorem subgrprop**
Step	Hyp	Ref	Expression
1		subgrv 16377	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2		issubgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝑆)
3		issubgr.a	. . . . 5 ⊢ 𝐴 = (Vtx‘𝐺)
4		issubgr.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝑆)
5		issubgr.b	. . . . 5 ⊢ 𝐵 = (iEdg‘𝐺)
6		issubgr.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝑆)
7	2, 3, 4, 5, 6	issubgr 16378	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
8	7	biimpd 144	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
9	8	ancoms 268	. 2 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
10	1, 9	mpcom 36	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 Vcvv 2815 ⊆ wss 3214 𝒫 cpw 3674 class class class wbr 4114 dom cdm 4754 ↾ cres 4756 ‘cfv 5357 Vtxcvtx 16133 iEdgciedg 16134 Edgcedg 16178 SubGraph csubgr 16374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-v 2817 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-xp 4760 df-rel 4761 df-dm 4764 df-res 4766 df-iota 5317 df-fv 5365 df-subgr 16375

This theorem is referenced by: subgrprop2 16381

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