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Theorem ssrest 22671
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 485 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐽t 𝐴))
2 ssrexv 4050 . . . . . 6 (𝐽𝐾 → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
32ad2antlr 725 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
4 n0i 4332 . . . . . . . 8 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
5 restfn 17366 . . . . . . . . . 10 t Fn (V × V)
65fndmi 6650 . . . . . . . . 9 dom ↾t = (V × V)
76ndmov 7587 . . . . . . . 8 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
84, 7nsyl2 141 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
98adantl 482 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
10 elrest 17369 . . . . . 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 765 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐾𝑉)
139simprd 496 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐴 ∈ V)
14 elrest 17369 . . . . . 6 ((𝐾𝑉𝐴 ∈ V) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
1512, 13, 14syl2anc 584 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
163, 11, 153imtr4d 293 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
171, 16mpd 15 . . 3 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐾t 𝐴))
1817ex 413 . 2 ((𝐾𝑉𝐽𝐾) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
1918ssrdv 3987 1 ((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  cin 3946  wss 3947  c0 4321   × cxp 5673  (class class class)co 7405  t crest 17362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-rest 17364
This theorem is referenced by:  1stcrest  22948  kgencmp  23040  kgencmp2  23041  kgen2ss  23050  ssufl  23413  cnfsmf  45442  smfsssmf  45445
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