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Theorem restutop 18259
 Description: Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
Assertion
Ref Expression
restutop UnifOn unifTopt unifTopt

Proof of Theorem restutop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . 4 UnifOn unifTopt UnifOn
2 fvex 5734 . . . . . . . 8 unifTop
32a1i 11 . . . . . . 7 UnifOn unifTop
4 elfvex 5750 . . . . . . . . 9 UnifOn
54adantr 452 . . . . . . . 8 UnifOn
6 simpr 448 . . . . . . . 8 UnifOn
75, 6ssexd 4342 . . . . . . 7 UnifOn
8 elrest 13647 . . . . . . 7 unifTop unifTopt unifTop
93, 7, 8syl2anc 643 . . . . . 6 UnifOn unifTopt unifTop
109biimpa 471 . . . . 5 UnifOn unifTopt unifTop
11 inss2 3554 . . . . . . 7
12 sseq1 3361 . . . . . . 7
1311, 12mpbiri 225 . . . . . 6
1413rexlimivw 2818 . . . . 5 unifTop
1510, 14syl 16 . . . 4 UnifOn unifTopt
16 simp-5l 745 . . . . . . . . . 10 UnifOn unifTopt unifTop UnifOn
1716ad2antrr 707 . . . . . . . . 9 UnifOn unifTopt unifTop UnifOn
187ad6antr 717 . . . . . . . . . 10 UnifOn unifTopt unifTop
19 xpexg 4981 . . . . . . . . . 10
2018, 18, 19syl2anc 643 . . . . . . . . 9 UnifOn unifTopt unifTop
21 simplr 732 . . . . . . . . 9 UnifOn unifTopt unifTop
22 elrestr 13648 . . . . . . . . 9 UnifOn t
2317, 20, 21, 22syl3anc 1184 . . . . . . . 8 UnifOn unifTopt unifTop t
24 inss1 3553 . . . . . . . . . . . . 13
25 imass1 5231 . . . . . . . . . . . . 13
2624, 25ax-mp 8 . . . . . . . . . . . 12
27 sstr 3348 . . . . . . . . . . . 12
2826, 27mpan 652 . . . . . . . . . . 11
29 imassrn 5208 . . . . . . . . . . . . . . 15
30 rnin 5273 . . . . . . . . . . . . . . 15
3129, 30sstri 3349 . . . . . . . . . . . . . 14
32 inss2 3554 . . . . . . . . . . . . . 14
3331, 32sstri 3349 . . . . . . . . . . . . 13
34 rnxpid 5294 . . . . . . . . . . . . 13
3533, 34sseqtri 3372 . . . . . . . . . . . 12
3635a1i 11 . . . . . . . . . . 11
3728, 36ssind 3557 . . . . . . . . . 10
3837adantl 453 . . . . . . . . 9 UnifOn unifTopt unifTop
39 simpllr 736 . . . . . . . . 9 UnifOn unifTopt unifTop
4038, 39sseqtr4d 3377 . . . . . . . 8 UnifOn unifTopt unifTop
41 imaeq1 5190 . . . . . . . . . 10
4241sseq1d 3367 . . . . . . . . 9
4342rspcev 3044 . . . . . . . 8 t t
4423, 40, 43syl2anc 643 . . . . . . 7 UnifOn unifTopt unifTop t
45 simplr 732 . . . . . . . 8 UnifOn unifTopt unifTop unifTop
46 inss1 3553 . . . . . . . . 9
47 simpllr 736 . . . . . . . . . 10 UnifOn unifTopt unifTop
48 simpr 448 . . . . . . . . . 10 UnifOn unifTopt unifTop
4947, 48eleqtrd 2511 . . . . . . . . 9 UnifOn unifTopt unifTop
5046, 49sseldi 3338 . . . . . . . 8 UnifOn unifTopt unifTop
51 elutop 18255 . . . . . . . . . 10 UnifOn unifTop
5251simplbda 608 . . . . . . . . 9 UnifOn unifTop
5352r19.21bi 2796 . . . . . . . 8 UnifOn unifTop
5416, 45, 50, 53syl21anc 1183 . . . . . . 7 UnifOn unifTopt unifTop
5544, 54r19.29a 2842 . . . . . 6 UnifOn unifTopt unifTop t
5610adantr 452 . . . . . 6 UnifOn unifTopt unifTop
5755, 56r19.29a 2842 . . . . 5 UnifOn unifTopt t
5857ralrimiva 2781 . . . 4 UnifOn unifTopt t
59 trust 18251 . . . . . 6 UnifOn t UnifOn
60 elutop 18255 . . . . . 6 t UnifOn unifTopt t
6159, 60syl 16 . . . . 5 UnifOn unifTopt t
6261biimpar 472 . . . 4 UnifOn t unifTopt
631, 15, 58, 62syl12anc 1182 . . 3 UnifOn unifTopt unifTopt
6463ex 424 . 2 UnifOn unifTopt unifTopt
6564ssrdv 3346 1 UnifOn unifTopt unifTopt
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948   cin 3311   wss 3312  csn 3806   cxp 4868   crn 4871  cima 4873  cfv 5446  (class class class)co 6073   ↾t crest 13640  UnifOncust 18221  unifTopcutop 18252 This theorem is referenced by:  restutopopn  18260 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-rest 13642  df-ust 18222  df-utop 18253
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