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Theorem isubgredgss 47851
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 ((𝐺𝑊𝑆𝑉) → 𝐼𝐸)

Proof of Theorem isubgredgss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isubgredg.h . . . . . 6 𝐻 = (𝐺 ISubGr 𝑆)
21fveq2i 6909 . . . . 5 (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆))
3 isubgredg.v . . . . . 6 𝑉 = (Vtx‘𝐺)
4 eqid 2737 . . . . . 6 (iEdg‘𝐺) = (iEdg‘𝐺)
53, 4isubgriedg 47849 . . . . 5 ((𝐺𝑊𝑆𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
62, 5eqtrid 2789 . . . 4 ((𝐺𝑊𝑆𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
76rneqd 5949 . . 3 ((𝐺𝑊𝑆𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
8 resss 6019 . . . 4 ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
9 rnss 5950 . . . 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))
108, 9mp1i 13 . . 3 ((𝐺𝑊𝑆𝑉) → ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ ran (iEdg‘𝐺))
117, 10eqsstrd 4018 . 2 ((𝐺𝑊𝑆𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺))
12 isubgredg.i . . 3 𝐼 = (Edg‘𝐻)
13 edgval 29066 . . 3 (Edg‘𝐻) = ran (iEdg‘𝐻)
1412, 13eqtri 2765 . 2 𝐼 = ran (iEdg‘𝐻)
15 isubgredg.e . . 3 𝐸 = (Edg‘𝐺)
16 edgval 29066 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
1715, 16eqtri 2765 . 2 𝐸 = ran (iEdg‘𝐺)
1811, 14, 173sstr4g 4037 1 ((𝐺𝑊𝑆𝑉) → 𝐼𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  wss 3951  dom cdm 5685  ran crn 5686  cres 5687  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064   ISubGr cisubgr 47846
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-iedg 29016  df-edg 29065  df-isubgr 47847
This theorem is referenced by: (None)
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