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Theorem ssrest 22073
Description: If 𝐾 is a finer topology than 𝐽, then the subspace topologies induced by 𝐴 maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
ssrest ((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))

Proof of Theorem ssrest
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 488 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐽t 𝐴))
2 ssrexv 3968 . . . . . 6 (𝐽𝐾 → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
32ad2antlr 727 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
4 n0i 4248 . . . . . . . 8 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
5 restfn 16929 . . . . . . . . . 10 t Fn (V × V)
65fndmi 6482 . . . . . . . . 9 dom ↾t = (V × V)
76ndmov 7392 . . . . . . . 8 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
84, 7nsyl2 143 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
98adantl 485 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
10 elrest 16932 . . . . . 6 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑥 = (𝑦𝐴)))
119, 10syl 17 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑥 = (𝑦𝐴)))
12 simpll 767 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐾𝑉)
139simprd 499 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐴 ∈ V)
14 elrest 16932 . . . . . 6 ((𝐾𝑉𝐴 ∈ V) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
1512, 13, 14syl2anc 587 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
163, 11, 153imtr4d 297 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
171, 16mpd 15 . . 3 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐾t 𝐴))
1817ex 416 . 2 ((𝐾𝑉𝐽𝐾) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
1918ssrdv 3907 1 ((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wrex 3062  Vcvv 3408  cin 3865  wss 3866  c0 4237   × cxp 5549  (class class class)co 7213  t crest 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-rest 16927
This theorem is referenced by:  1stcrest  22350  kgencmp  22442  kgencmp2  22443  kgen2ss  22452  ssufl  22815  cnfsmf  43948  smfsssmf  43951
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