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Theorem ssrest 20893
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 477 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐽t 𝐴))
2 ssrexv 3648 . . . . . 6 (𝐽𝐾 → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
32ad2antlr 762 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
4 n0i 3898 . . . . . . . 8 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
5 restfn 16009 . . . . . . . . . 10 t Fn (V × V)
6 fndm 5950 . . . . . . . . . 10 ( ↾t Fn (V × V) → dom ↾t = (V × V))
75, 6ax-mp 5 . . . . . . . . 9 dom ↾t = (V × V)
87ndmov 6774 . . . . . . . 8 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
94, 8nsyl2 142 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
109adantl 482 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
11 elrest 16012 . . . . . 6 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑥 = (𝑦𝐴)))
1210, 11syl 17 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑥 = (𝑦𝐴)))
13 simpll 789 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐾𝑉)
1410simprd 479 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐴 ∈ V)
15 elrest 16012 . . . . . 6 ((𝐾𝑉𝐴 ∈ V) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
1613, 14, 15syl2anc 692 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
173, 12, 163imtr4d 283 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
181, 17mpd 15 . . 3 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐾t 𝐴))
1918ex 450 . 2 ((𝐾𝑉𝐽𝐾) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
2019ssrdv 3590 1 ((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  cin 3555  wss 3556  c0 3893   × cxp 5074  dom cdm 5076   Fn wfn 5844  (class class class)co 6607  t crest 16005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-rest 16007
This theorem is referenced by:  1stcrest  21169  kgencmp  21261  kgencmp2  21262  kgen2ss  21271  ssufl  21635  cnfsmf  40272  smfsssmf  40275
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