Theorem isubgredgss 48492

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 6872	. . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3		isubgredg.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4		eqid 2764	. . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5	3, 4	isubgriedg 48490	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
6	2, 5	eqtrid 2811	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
7	6	rneqd 5916	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
8		resss 5989	. . . 4 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
9		rnss 5917	. . . 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 3972	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺))
12		isubgredg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐻)
13		edgval 29252	. . 3 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻)
14	12, 13	eqtri 2787	. 2 ⊢ 𝐼 = ran (iEdg‘𝐻)
15		isubgredg.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
16		edgval 29252	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
17	15, 16	eqtri 2787	. 2 ⊢ 𝐸 = ran (iEdg‘𝐺)
18	11, 14, 17	3sstr4g 3991	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {crab 3416   ⊆ wss 3906  dom cdm 5649  ran crn 5650   ↾ cres 5651  ‘cfv 6523  (class class class)co 7398  Vtxcvtx 29199  iEdgciedg 29200  Edgcedg 29250   ISubGr cisubgr 48487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-2nd 7973  df-iedg 29202  df-edg 29251  df-isubgr 48488

This theorem is referenced by: (None)