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Theorem subgrprop 16109

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 16106	. 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 16107	. . . 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 1004 = wceq 1397 ∈ wcel 2202 Vcvv 2802 ⊆ wss 3200 𝒫 cpw 3652 class class class wbr 4088 dom cdm 4725 ↾ cres 4727 ‘cfv 5326 Vtxcvtx 15862 iEdgciedg 15863 Edgcedg 15907 SubGraph csubgr 16103

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-xp 4731 df-rel 4732 df-dm 4735 df-res 4737 df-iota 5286 df-fv 5334 df-subgr 16104

This theorem is referenced by: subgrprop2 16110

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