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Theorem restsspw 17386
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 4328 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
2 restfn 17379 . . . . . . . . 9 t Fn (V × V)
3 fndm 6646 . . . . . . . . 9 ( ↾t Fn (V × V) → dom ↾t = (V × V))
42, 3ax-mp 5 . . . . . . . 8 dom ↾t = (V × V)
54ndmov 7588 . . . . . . 7 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
61, 5nsyl2 141 . . . . . 6 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
7 elrest 17382 . . . . . 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 4224 . . . . . 6 (𝑦𝐴) ⊆ 𝐴
11 sseq1 4002 . . . . . 6 (𝑥 = (𝑦𝐴) → (𝑥𝐴 ↔ (𝑦𝐴) ⊆ 𝐴))
1210, 11mpbiri 258 . . . . 5 (𝑥 = (𝑦𝐴) → 𝑥𝐴)
1312rexlimivw 3145 . . . 4 (∃𝑦𝐽 𝑥 = (𝑦𝐴) → 𝑥𝐴)
149, 13syl 17 . . 3 (𝑥 ∈ (𝐽t 𝐴) → 𝑥𝐴)
15 velpw 4602 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1614, 15sylibr 233 . 2 (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ 𝒫 𝐴)
1716ssriv 3981 1 (𝐽t 𝐴) ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  cin 3942  wss 3943  c0 4317  𝒫 cpw 4597   × cxp 5667  dom cdm 5669   Fn wfn 6532  (class class class)co 7405  t crest 17375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-rest 17377
This theorem is referenced by:  1stckgenlem  23412  prdstopn  23487  trfbas2  23702  trfil1  23745  trfil2  23746  fgtr  23749  trust  24089  zdis  24687  cnambfre  37049  dvdmsscn  45224  dvnmptconst  45229  dvnxpaek  45230  dvnmul  45231  dvnprodlem3  45236
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