Theorem isubgredgss 47869

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 6864	. . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3		isubgredg.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4		eqid 2730	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	3, 4	isubgriedg 47867	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
6	2, 5	eqtrid 2777	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
7	6	rneqd 5905	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
8		resss 5975	. . . 4 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
9		rnss 5906	. . . 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 3984	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺))
12		isubgredg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐻)
13		edgval 28983	. . 3 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻)
14	12, 13	eqtri 2753	. 2 ⊢ 𝐼 = ran (iEdg‘𝐻)
15		isubgredg.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
16		edgval 28983	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
17	15, 16	eqtri 2753	. 2 ⊢ 𝐸 = ran (iEdg‘𝐺)
18	11, 14, 17	3sstr4g 4003	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408   ⊆ wss 3917  dom cdm 5641  ran crn 5642   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  iEdgciedg 28931  Edgcedg 28981   ISubGr cisubgr 47864

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-iedg 28933  df-edg 28982  df-isubgr 47865

This theorem is referenced by: (None)