Theorem subgrprop3 16257

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 2232	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2232	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		subgrprop3.e	. . . 4 ⊢ 𝐸 = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 16255	. . 3 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7		3simpa 1021	. . 3 ⊢ ((𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
8	6, 7	syl 14	. 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
9		simprl 531	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉 ⊆ 𝐴)
10		rnss 4987	. . . . 5 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
11	10	ad2antll 491	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12		subgrv 16251	. . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13		edgval 16055	. . . . . . . . 9 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
14	13	a1i 9	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
15	5, 14	eqtrid 2277	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16		subgrprop3.b	. . . . . . . 8 ⊢ 𝐵 = (Edg‘𝐺)
17		edgval 16055	. . . . . . . . 9 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
18	17	a1i 9	. . . . . . . 8 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
19	16, 18	eqtrid 2277	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
20	15, 19	sseq12d 3269	. . . . . 6 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
21	12, 20	syl 14	. . . . 5 ⊢ (𝑆 SubGraph 𝐺 → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
22	21	adantr 276	. . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸 ⊆ 𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
23	11, 22	mpbird 167	. . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸 ⊆ 𝐵)
24	9, 23	jca 306	. 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ (𝑉 ⊆ 𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))
25	8, 24	mpdan 421	1 ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  Vcvv 2813   ⊆ wss 3211  𝒫 cpw 3669   class class class wbr 4109  ran crn 4750  ‘cfv 5352  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052   SubGraph csubgr 16248

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-2nd 6335  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-edgf 16000  df-iedg 16010  df-edg 16053  df-subgr 16249

This theorem is referenced by: (None)