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Theorem isisubgr 47848
Description: The subgraph induced by a subset of vertices. (Contributed by AV, 12-May-2025.)
Hypotheses
Ref Expression
isisubgr.v 𝑉 = (Vtx‘𝐺)
isisubgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isisubgr ((𝐺𝑊𝑆𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩)
Distinct variable groups:   𝑥,𝐸   𝑥,𝐺   𝑥,𝑆   𝑥,𝑉
Allowed substitution hint:   𝑊(𝑥)

Proof of Theorem isisubgr
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3501 . . 3 (𝐺𝑊𝐺 ∈ V)
21adantr 480 . 2 ((𝐺𝑊𝑆𝑉) → 𝐺 ∈ V)
3 isisubgr.v . . . . . 6 𝑉 = (Vtx‘𝐺)
43fvexi 6920 . . . . 5 𝑉 ∈ V
54a1i 11 . . . 4 (𝑆𝑉𝑉 ∈ V)
6 id 22 . . . 4 (𝑆𝑉𝑆𝑉)
75, 6sselpwd 5328 . . 3 (𝑆𝑉𝑆 ∈ 𝒫 𝑉)
87adantl 481 . 2 ((𝐺𝑊𝑆𝑉) → 𝑆 ∈ 𝒫 𝑉)
9 opex 5469 . . 3 𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩ ∈ V
109a1i 11 . 2 ((𝐺𝑊𝑆𝑉) → ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩ ∈ V)
11 simpr 484 . . . 4 ((𝑔 = 𝐺𝑣 = 𝑆) → 𝑣 = 𝑆)
12 fvexd 6921 . . . . 5 ((𝑔 = 𝐺𝑣 = 𝑆) → (iEdg‘𝑔) ∈ V)
13 fveq2 6906 . . . . . . . . . 10 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
14 isisubgr.e . . . . . . . . . 10 𝐸 = (iEdg‘𝐺)
1513, 14eqtr4di 2795 . . . . . . . . 9 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐸)
1615eqeq2d 2748 . . . . . . . 8 (𝑔 = 𝐺 → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸))
1716adantr 480 . . . . . . 7 ((𝑔 = 𝐺𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸))
18 simpr 484 . . . . . . . . . 10 ((𝑣 = 𝑆𝑒 = 𝐸) → 𝑒 = 𝐸)
19 dmeq 5914 . . . . . . . . . . . 12 (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
2019adantl 481 . . . . . . . . . . 11 ((𝑣 = 𝑆𝑒 = 𝐸) → dom 𝑒 = dom 𝐸)
21 fveq1 6905 . . . . . . . . . . . . 13 (𝑒 = 𝐸 → (𝑒𝑥) = (𝐸𝑥))
2221adantl 481 . . . . . . . . . . . 12 ((𝑣 = 𝑆𝑒 = 𝐸) → (𝑒𝑥) = (𝐸𝑥))
23 simpl 482 . . . . . . . . . . . 12 ((𝑣 = 𝑆𝑒 = 𝐸) → 𝑣 = 𝑆)
2422, 23sseq12d 4017 . . . . . . . . . . 11 ((𝑣 = 𝑆𝑒 = 𝐸) → ((𝑒𝑥) ⊆ 𝑣 ↔ (𝐸𝑥) ⊆ 𝑆))
2520, 24rabeqbidv 3455 . . . . . . . . . 10 ((𝑣 = 𝑆𝑒 = 𝐸) → {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣} = {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})
2618, 25reseq12d 5998 . . . . . . . . 9 ((𝑣 = 𝑆𝑒 = 𝐸) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆}))
2726ex 412 . . . . . . . 8 (𝑣 = 𝑆 → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})))
2827adantl 481 . . . . . . 7 ((𝑔 = 𝐺𝑣 = 𝑆) → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})))
2917, 28sylbid 240 . . . . . 6 ((𝑔 = 𝐺𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})))
3029imp 406 . . . . 5 (((𝑔 = 𝐺𝑣 = 𝑆) ∧ 𝑒 = (iEdg‘𝑔)) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆}))
3112, 30csbied 3935 . . . 4 ((𝑔 = 𝐺𝑣 = 𝑆) → (iEdg‘𝑔) / 𝑒(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆}))
3211, 31opeq12d 4881 . . 3 ((𝑔 = 𝐺𝑣 = 𝑆) → ⟨𝑣, (iEdg‘𝑔) / 𝑒(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣})⟩ = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩)
33 fveq2 6906 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3433, 3eqtr4di 2795 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
3534pweqd 4617 . . 3 (𝑔 = 𝐺 → 𝒫 (Vtx‘𝑔) = 𝒫 𝑉)
36 df-isubgr 47847 . . 3 ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, (iEdg‘𝑔) / 𝑒(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) ⊆ 𝑣})⟩)
3732, 35, 36ovmpox 7586 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉 ∧ ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩ ∈ V) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩)
382, 8, 10, 37syl3anc 1373 1 ((𝐺𝑊𝑆𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑆})⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  csb 3899  wss 3951  𝒫 cpw 4600  cop 4632  dom cdm 5685  cres 5687  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014   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
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-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-isubgr 47847
This theorem is referenced by:  isubgriedg  47849  isubgrvtxuhgr  47850  isubgrvtx  47853  isubgr0uhgr  47859
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