Theorem isubgredgss 48053

Description: The edges of an induced subgraph of a graph are edges of the graph. (Contributed by AV, 24-Sep-2025.)

Hypotheses

Ref	Expression
isubgredg.v	⊢ 𝑉 = (Vtx‘𝐺)
isubgredg.e	⊢ 𝐸 = (Edg‘𝐺)
isubgredg.h	⊢ 𝐻 = (𝐺 ISubGr 𝑆)
isubgredg.i	⊢ 𝐼 = (Edg‘𝐻)

Assertion

Ref	Expression
isubgredgss	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸)

Proof of Theorem isubgredgss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgredg.h	. . . . . 6 ⊢ 𝐻 = (𝐺 ISubGr 𝑆)
2	1	fveq2i 6835	. . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3		isubgredg.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4		eqid 2734	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	3, 4	isubgriedg 48051	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
6	2, 5	eqtrid 2781	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
7	6	rneqd 5885	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
8		resss 5958	. . . 4 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
9		rnss 5886	. . . 4 ⊢ (((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺) → ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ ran (iEdg‘𝐺))
10	8, 9	mp1i 13	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ ran (iEdg‘𝐺))
11	7, 10	eqsstrd 3966	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺))
12		isubgredg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐻)
13		edgval 29071	. . 3 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻)
14	12, 13	eqtri 2757	. 2 ⊢ 𝐼 = ran (iEdg‘𝐻)
15		isubgredg.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
16		edgval 29071	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
17	15, 16	eqtri 2757	. 2 ⊢ 𝐸 = ran (iEdg‘𝐺)
18	11, 14, 17	3sstr4g 3985	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3397   ⊆ wss 3899  dom cdm 5622  ran crn 5623   ↾ cres 5624  ‘cfv 6490  (class class class)co 7356  Vtxcvtx 29018  iEdgciedg 29019  Edgcedg 29069   ISubGr cisubgr 48048

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-2nd 7932  df-iedg 29021  df-edg 29070  df-isubgr 48049

This theorem is referenced by: (None)