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Theorem restlly 17551
 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 3350 . . . 4
3 simprl 734 . . . . . . 7
4 vex 2961 . . . . . . . . 9
54pwid 3814 . . . . . . . 8
65a1i 11 . . . . . . 7
7 elin 3532 . . . . . . 7
83, 6, 7sylanbrc 647 . . . . . 6
9 simprr 735 . . . . . 6
10 restlly.1 . . . . . . . 8 t
1110anassrs 631 . . . . . . 7 t
1211adantrr 699 . . . . . 6 t
13 eleq2 2499 . . . . . . . 8
14 oveq2 6092 . . . . . . . . 9 t t
1514eleq1d 2504 . . . . . . . 8 t t
1613, 15anbi12d 693 . . . . . . 7 t t
1716rspcev 3054 . . . . . 6 t t
188, 9, 12, 17syl12anc 1183 . . . . 5 t
1918ralrimivva 2800 . . . 4 t
20 islly 17536 . . . 4 Locally t
212, 19, 20sylanbrc 647 . . 3 Locally
2221ex 425 . 2 Locally
2322ssrdv 3356 1 Locally
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wcel 1726  wral 2707  wrex 2708   cin 3321   wss 3322  cpw 3801  (class class class)co 6084   ↾t crest 13653  ctop 16963  Locally clly 17532 This theorem is referenced by:  llyidm  17556  nllyidm  17557  toplly  17558  hauslly  17560  lly1stc  17564 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087  df-lly 17534
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