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Theorem ssrest 23199
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 484 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐽t 𝐴))
2 ssrexv 4064 . . . . . 6 (𝐽𝐾 → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
32ad2antlr 727 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
4 n0i 4345 . . . . . . . 8 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
5 restfn 17470 . . . . . . . . . 10 t Fn (V × V)
65fndmi 6672 . . . . . . . . 9 dom ↾t = (V × V)
76ndmov 7616 . . . . . . . 8 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
84, 7nsyl2 141 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
98adantl 481 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
10 elrest 17473 . . . . . 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 495 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐴 ∈ V)
14 elrest 17473 . . . . . 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 294 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
171, 16mpd 15 . . 3 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐾t 𝐴))
1817ex 412 . 2 ((𝐾𝑉𝐽𝐾) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
1918ssrdv 4000 1 ((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  cin 3961  wss 3962  c0 4338   × cxp 5686  (class class class)co 7430  t crest 17466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-rest 17468
This theorem is referenced by:  1stcrest  23476  kgencmp  23568  kgencmp2  23569  kgen2ss  23578  ssufl  23941  cnfsmf  46695  smfsssmf  46698
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