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Theorem restsspw 17446
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 4336 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
2 restfn 17439 . . . . . . . . 9 t Fn (V × V)
3 fndm 6663 . . . . . . . . 9 ( ↾t Fn (V × V) → dom ↾t = (V × V))
42, 3ax-mp 5 . . . . . . . 8 dom ↾t = (V × V)
54ndmov 7610 . . . . . . 7 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
61, 5nsyl2 141 . . . . . 6 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
7 elrest 17442 . . . . . 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 266 . . . 4 (𝑥 ∈ (𝐽t 𝐴) → ∃𝑦𝐽 𝑥 = (𝑦𝐴))
10 inss2 4231 . . . . . 6 (𝑦𝐴) ⊆ 𝐴
11 sseq1 4005 . . . . . 6 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1210, 11mpbiri 257 . . . . 5 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1312rexlimivw 3141 . . . 4 (∃𝑦𝐽 𝑥 = (𝑦𝐴) → 𝑥𝐴)
149, 13syl 17 . . 3 (𝑥 ∈ (𝐽t 𝐴) → 𝑥𝐴)
15 velpw 4612 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1614, 15sylibr 233 . 2 (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ 𝒫 𝐴)
1716ssriv 3983 1 (𝐽t 𝐴) ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  Vcvv 3462  cin 3946  wss 3947  c0 4325  𝒫 cpw 4607   × cxp 5680  dom cdm 5682   Fn wfn 6549  (class class class)co 7424  t crest 17435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-rest 17437
This theorem is referenced by:  1stckgenlem  23548  prdstopn  23623  trfbas2  23838  trfil1  23881  trfil2  23882  fgtr  23885  trust  24225  zdis  24823  cnambfre  37369  dvdmsscn  45557  dvnmptconst  45562  dvnxpaek  45563  dvnmul  45564  dvnprodlem3  45569
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