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Theorem isubgredgss 47788
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 2734 . . . . . 6 (iEdg‘𝐺) = (iEdg‘𝐺)
53, 4isubgriedg 47786 . . . . 5 ((𝐺𝑊𝑆𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
62, 5eqtrid 2786 . . . 4 ((𝐺𝑊𝑆𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
76rneqd 5951 . . 3 ((𝐺𝑊𝑆𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
8 resss 6021 . . . 4 ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺)
9 rnss 5952 . . . 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 4033 . 2 ((𝐺𝑊𝑆𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺))
12 isubgredg.i . . 3 𝐼 = (Edg‘𝐻)
13 edgval 29080 . . 3 (Edg‘𝐻) = ran (iEdg‘𝐻)
1412, 13eqtri 2762 . 2 𝐼 = ran (iEdg‘𝐻)
15 isubgredg.e . . 3 𝐸 = (Edg‘𝐺)
16 edgval 29080 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
1715, 16eqtri 2762 . 2 𝐸 = ran (iEdg‘𝐺)
1811, 14, 173sstr4g 4040 1 ((𝐺𝑊𝑆𝑉) → 𝐼𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {crab 3432  wss 3962  dom cdm 5688  ran crn 5689  cres 5690  cfv 6562  (class class class)co 7430  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078   ISubGr cisubgr 47783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-2nd 8013  df-iedg 29030  df-edg 29079  df-isubgr 47784
This theorem is referenced by: (None)
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