Theorem subgrprop3 27931

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. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
subgrprop3.v	⊢ 𝑉 = (Vtx‘𝑆)
subgrprop3.a	⊢ 𝐴 = (Vtx‘𝐺)
subgrprop3.e	⊢ 𝐸 = (Edg‘𝑆)
subgrprop3.b	⊢ 𝐵 = (Edg‘𝐺)

Assertion

Ref	Expression
subgrprop3	⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))

Proof of Theorem subgrprop3

Step	Hyp	Ref	Expression
1		subgrprop3.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝑆)
2		subgrprop3.a	. . . 4 ⊢ 𝐴 = (Vtx‘𝐺)
3		eqid 2737	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2737	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		subgrprop3.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 27929	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		3simpa 1148	. . 3 ⊢ ((𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
8	6, 7	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
9		simprl 769	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉 ⊆ 𝐴)
10		rnss 5884	. . . . 5 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
11	10	ad2antll 727	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12		subgrv 27925	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13		edgval 27707	. . . . . . . . 9 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
15	5, 14	eqtrid 2789	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16		subgrprop3.b	. . . . . . . 8 ⊢ 𝐵 = (Edg‘𝐺)
17		edgval 27707	. . . . . . . . 9 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
19	16, 18	eqtrid 2789	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
20	15, 19	sseq12d 3968	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
21	12, 20	syl 17	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
22	21	adantr 482	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
23	11, 22	mpbird 257	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸 ⊆ 𝐵)
24	9, 23	jca 513	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))
25	8, 24	mpdan 685	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3442   ⊆ wss 3901  𝒫 cpw 4551   class class class wbr 5096  ran crn 5625  ‘cfv 6483  Vtxcvtx 27654  iEdgciedg 27655  Edgcedg 27705   SubGraph csubgr 27922

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376  ax-un 7654

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6435  df-fun 6485  df-fv 6491  df-edg 27706  df-subgr 27923

This theorem is referenced by: (None)