Theorem subgrprop3 29249

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 2731	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2731	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		subgrprop3.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29247	. . 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 770	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉 ⊆ 𝐴)
10		rnss 5874	. . . . 5 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
11	10	ad2antll 729	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12		subgrv 29243	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13		edgval 29022	. . . . . . . . 9 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
15	5, 14	eqtrid 2778	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16		subgrprop3.b	. . . . . . . 8 ⊢ 𝐵 = (Edg‘𝐺)
17		edgval 29022	. . . . . . . . 9 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
19	16, 18	eqtrid 2778	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
20	15, 19	sseq12d 3963	. . . . . 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 480	. . . 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 511	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))
25	8, 24	mpdan 687	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4545   class class class wbr 5086  ran crn 5612  ‘cfv 6476  Vtxcvtx 28969  iEdgciedg 28970  Edgcedg 29020   SubGraph csubgr 29240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-iota 6432  df-fun 6478  df-fv 6484  df-edg 29021  df-subgr 29241

This theorem is referenced by: (None)