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Theorem ucnprima 21228
 Description: The preimage by a uniformly continuous function of an entourage of is an entourage of . Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1 UnifOn
ucnprima.2 UnifOn
ucnprima.3 Cnu
ucnprima.4
ucnprima.5
Assertion
Ref Expression
ucnprima
Distinct variable groups:   ,,   ,,   ,,   ,,   ,   ,,   ,   ,,
Allowed substitution hints:   ()   ()

Proof of Theorem ucnprima
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ucnprima.1 . . . 4 UnifOn
2 ucnprima.2 . . . 4 UnifOn
3 ucnprima.3 . . . 4 Cnu
4 ucnprima.4 . . . 4
5 ucnprima.5 . . . 4
61, 2, 3, 4, 5ucnima 21227 . . 3
75mpt2fun 6412 . . . . 5
8 ustssxp 21150 . . . . . . 7 UnifOn
91, 8sylan 473 . . . . . 6
10 opex 4686 . . . . . . 7
115, 10dmmpt2 6877 . . . . . 6
129, 11syl6sseqr 3517 . . . . 5
13 funimass3 6013 . . . . 5
147, 12, 13sylancr 667 . . . 4
1514rexbidva 2943 . . 3
166, 15mpbid 213 . 2
171adantr 466 . . . 4 UnifOn
18 simpr 462 . . . 4
19 cnvimass 5208 . . . . . 6
2019, 11sseqtri 3502 . . . . 5
2120a1i 11 . . . 4
22 ustssel 21151 . . . 4 UnifOn
2317, 18, 21, 22syl3anc 1264 . . 3
2423rexlimdva 2924 . 2
2516, 24mpd 15 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1870  wrex 2783   wss 3442  cop 4008   cxp 4852  ccnv 4853   cdm 4854  cima 4857   wfun 5595  cfv 5601  (class class class)co 6305   cmpt2 6307  UnifOncust 21145   Cnucucn 21221 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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-ust 21146  df-ucn 21222 This theorem is referenced by:  fmucnd  21238
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