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Theorem restsspw 17476
Description: The subspace topology is a collection of subsets of the restriction set. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
restsspw (𝐽t 𝐴) ⊆ 𝒫 𝐴

Proof of Theorem restsspw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4340 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
2 restfn 17469 . . . . . . . . 9 t Fn (V × V)
3 fndm 6671 . . . . . . . . 9 ( ↾t Fn (V × V) → dom ↾t = (V × V))
42, 3ax-mp 5 . . . . . . . 8 dom ↾t = (V × V)
54ndmov 7617 . . . . . . 7 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
61, 5nsyl2 141 . . . . . 6 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
7 elrest 17472 . . . . . 6 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑥 = (𝑦𝐴)))
86, 7syl 17 . . . . 5 (𝑥 ∈ (𝐽t 𝐴) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑥 = (𝑦𝐴)))
98ibi 267 . . . 4 (𝑥 ∈ (𝐽t 𝐴) → ∃𝑦𝐽 𝑥 = (𝑦𝐴))
10 inss2 4238 . . . . . 6 (𝑦𝐴) ⊆ 𝐴
11 sseq1 4009 . . . . . 6 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1210, 11mpbiri 258 . . . . 5 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1312rexlimivw 3151 . . . 4 (∃𝑦𝐽 𝑥 = (𝑦𝐴) → 𝑥𝐴)
149, 13syl 17 . . 3 (𝑥 ∈ (𝐽t 𝐴) → 𝑥𝐴)
15 velpw 4605 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1614, 15sylibr 234 . 2 (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ 𝒫 𝐴)
1716ssriv 3987 1 (𝐽t 𝐴) ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   × cxp 5683  dom cdm 5685   Fn wfn 6556  (class class class)co 7431  t crest 17465
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-rep 5279  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-reu 3381  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-iun 4993  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-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467
This theorem is referenced by:  1stckgenlem  23561  prdstopn  23636  trfbas2  23851  trfil1  23894  trfil2  23895  fgtr  23898  trust  24238  zdis  24838  cnambfre  37675  dvdmsscn  45951  dvnmptconst  45956  dvnxpaek  45957  dvnmul  45958  dvnprodlem3  45963
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