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

Theorem isusp 20632
 Description: The predicate is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
isusp.1
isusp.2 UnifSt
isusp.3
Assertion
Ref Expression
isusp UnifSp UnifOn unifTop

Proof of Theorem isusp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2 UnifSp
2 0nep0 4624 . . . . 5
3 isusp.1 . . . . . . . . . . . 12
4 fvprc 5866 . . . . . . . . . . . 12
53, 4syl5eq 2520 . . . . . . . . . . 11
65fveq2d 5876 . . . . . . . . . 10 UnifOn UnifOn
7 ust0 20590 . . . . . . . . . 10 UnifOn
86, 7syl6eq 2524 . . . . . . . . 9 UnifOn
98eleq2d 2537 . . . . . . . 8 UnifOn
10 isusp.2 . . . . . . . . . 10 UnifSt
11 fvex 5882 . . . . . . . . . 10 UnifSt
1210, 11eqeltri 2551 . . . . . . . . 9
1312elsnc 4057 . . . . . . . 8
149, 13syl6bb 261 . . . . . . 7 UnifOn
15 fvprc 5866 . . . . . . . . 9 UnifSt
1610, 15syl5eq 2520 . . . . . . . 8
1716eqeq1d 2469 . . . . . . 7
1814, 17bitrd 253 . . . . . 6 UnifOn
1918necon3bbid 2714 . . . . 5 UnifOn
202, 19mpbiri 233 . . . 4 UnifOn
2120con4i 130 . . 3 UnifOn
2221adantr 465 . 2 UnifOn unifTop
23 fveq2 5872 . . . . . 6 UnifSt UnifSt
2423, 10syl6eqr 2526 . . . . 5 UnifSt
25 fveq2 5872 . . . . . . 7
2625, 3syl6eqr 2526 . . . . . 6
2726fveq2d 5876 . . . . 5 UnifOn UnifOn
2824, 27eleq12d 2549 . . . 4 UnifSt UnifOn UnifOn
29 fveq2 5872 . . . . . 6
30 isusp.3 . . . . . 6
3129, 30syl6eqr 2526 . . . . 5
3224fveq2d 5876 . . . . 5 unifTopUnifSt unifTop
3331, 32eqeq12d 2489 . . . 4 unifTopUnifSt unifTop
3428, 33anbi12d 710 . . 3 UnifSt UnifOn unifTopUnifSt UnifOn unifTop
35 df-usp 20628 . . 3 UnifSp UnifSt UnifOn unifTopUnifSt
3634, 35elab2g 3257 . 2 UnifSp UnifOn unifTop
371, 22, 36pm5.21nii 353 1 UnifSp UnifOn unifTop
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 184   wa 369   wceq 1379   wcel 1767   wne 2662  cvv 3118  c0 3790  csn 4033  cfv 5594  cbs 14507  ctopn 14694  UnifOncust 20570  unifTopcutop 20601  UnifStcuss 20624  UnifSpcusp 20625 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-ust 20571  df-usp 20628 This theorem is referenced by:  ressust  20635  ressusp  20636  tususp  20643  uspreg  20645  ucncn  20656  neipcfilu  20667  ucnextcn  20675  xmsuspOLD  20956  xmsusp  20957
 Copyright terms: Public domain W3C validator