Theorem subgrprop3 29260

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 2736	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		subgrprop3.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29258	. . 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 5924	. . . . 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 29254	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13		edgval 29033	. . . . . . . . 9 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
15	5, 14	eqtrid 2783	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16		subgrprop3.b	. . . . . . . 8 ⊢ 𝐵 = (Edg‘𝐺)
17		edgval 29033	. . . . . . . . 9 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
19	16, 18	eqtrid 2783	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
20	15, 19	sseq12d 3997	. . . . . 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 1540   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  𝒫 cpw 4580   class class class wbr 5124  ran crn 5660  ‘cfv 6536  Vtxcvtx 28980  iEdgciedg 28981  Edgcedg 29031   SubGraph csubgr 29251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6538  df-fv 6544  df-edg 29032  df-subgr 29252

This theorem is referenced by: (None)