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Theorem ssrest 22902
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 483 . . . 4 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝑥 ∈ (𝐽t 𝐴))
2 ssrexv 4052 . . . . . 6 (𝐽𝐾 → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
32ad2antlr 723 . . . . 5 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (∃𝑦𝐽 𝑥 = (𝑦𝐴) → ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
4 n0i 4334 . . . . . . . 8 (𝑥 ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
5 restfn 17376 . . . . . . . . . 10 t Fn (V × V)
65fndmi 6654 . . . . . . . . 9 dom ↾t = (V × V)
76ndmov 7595 . . . . . . . 8 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
84, 7nsyl2 141 . . . . . . 7 (𝑥 ∈ (𝐽t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
98adantl 480 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V))
10 elrest 17379 . . . . . 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 763 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐾𝑉)
139simprd 494 . . . . . 6 (((𝐾𝑉𝐽𝐾) ∧ 𝑥 ∈ (𝐽t 𝐴)) → 𝐴 ∈ V)
14 elrest 17379 . . . . . 6 ((𝐾𝑉𝐴 ∈ V) → (𝑥 ∈ (𝐾t 𝐴) ↔ ∃𝑦𝐾 𝑥 = (𝑦𝐴)))
1512, 13, 14syl2anc 582 . . . . 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 411 . 2 ((𝐾𝑉𝐽𝐾) → (𝑥 ∈ (𝐽t 𝐴) → 𝑥 ∈ (𝐾t 𝐴)))
1918ssrdv 3989 1 ((𝐾𝑉𝐽𝐾) → (𝐽t 𝐴) ⊆ (𝐾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wrex 3068  Vcvv 3472  cin 3948  wss 3949  c0 4323   × cxp 5675  (class class class)co 7413  t crest 17372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7979  df-2nd 7980  df-rest 17374
This theorem is referenced by:  1stcrest  23179  kgencmp  23271  kgencmp2  23272  kgen2ss  23281  ssufl  23644  cnfsmf  45756  smfsssmf  45759
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