Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  restlly Structured version   Unicode version

Theorem restlly 20484
 Description: If the property passes to open subspaces, then a space which is is also locally . (Contributed by Mario Carneiro, 2-Mar-2015.)
Hypotheses
Ref Expression
restlly.1 t
restlly.2
Assertion
Ref Expression
restlly Locally
Distinct variable groups:   ,,   ,,

Proof of Theorem restlly
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 restlly.2 . . . . 5
21sselda 3464 . . . 4
3 simprl 762 . . . . . . 7
4 vex 3084 . . . . . . . . 9
54pwid 3993 . . . . . . . 8
65a1i 11 . . . . . . 7
73, 6elind 3650 . . . . . 6
8 simprr 764 . . . . . 6
9 restlly.1 . . . . . . . 8 t
109anassrs 652 . . . . . . 7 t
1110adantrr 721 . . . . . 6 t
12 eleq2 2495 . . . . . . . 8
13 oveq2 6309 . . . . . . . . 9 t t
1413eleq1d 2491 . . . . . . . 8 t t
1512, 14anbi12d 715 . . . . . . 7 t t
1615rspcev 3182 . . . . . 6 t t
177, 8, 11, 16syl12anc 1262 . . . . 5 t
1817ralrimivva 2846 . . . 4 t
19 islly 20469 . . . 4 Locally t
202, 18, 19sylanbrc 668 . . 3 Locally
2120ex 435 . 2 Locally
2221ssrdv 3470 1 Locally
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wcel 1868  wral 2775  wrex 2776   cin 3435   wss 3436  cpw 3979  (class class class)co 6301   ↾t crest 15306  ctop 19903  Locally clly 20465 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-iota 5561  df-fv 5605  df-ov 6304  df-lly 20467 This theorem is referenced by:  llyidm  20489  nllyidm  20490  toplly  20491  hauslly  20493  lly1stc  20497
 Copyright terms: Public domain W3C validator