Theorem subgrprop3 29872

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 2763	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		subgrprop3.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 29870	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		3simpa 1166	. . 3 ⊢ ((𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
8	6, 7	syl 18	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
9		simprl 783	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉 ⊆ 𝐴)
10		rnss 5926	. . . . 5 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
11	10	ad2antll 742	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12		subgrv 29866	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13		edgval 29642	. . . . . . . . 9 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
15	5, 14	eqtrid 2810	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16		subgrprop3.b	. . . . . . . 8 ⊢ 𝐵 = (Edg‘𝐺)
17		edgval 29642	. . . . . . . . 9 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
19	16, 18	eqtrid 2810	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
20	15, 19	sseq12d 3946	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
21	12, 20	syl 18	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
22	21	adantr 486	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
23	11, 22	mpbird 260	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸 ⊆ 𝐵)
24	9, 23	jca 521	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))
25	8, 24	mpdan 700	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5106  ran crn 5656  ‘cfv 6547  Vtxcvtx 29589  iEdgciedg 29590  Edgcedg 29640   SubGraph csubgr 29863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6501  df-fun 6549  df-fv 6555  df-edg 29641  df-subgr 29864

This theorem is used by: (None)