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Theorem ucnval 21370
 Description: The set of all uniformly continuous function from uniform space to uniform space . (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ucnval UnifOn UnifOn Cnu
Distinct variable groups:   ,,,,,   ,,,,   ,,,,,   ,,,,
Allowed substitution hints:   ()   ()

Proof of Theorem ucnval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrnust 21317 . . . 4 UnifOn UnifOn
21adantr 472 . . 3 UnifOn UnifOn UnifOn
3 elrnust 21317 . . . 4 UnifOn UnifOn
43adantl 473 . . 3 UnifOn UnifOn UnifOn
5 ovex 6336 . . . . 5
65rabex 4550 . . . 4
76a1i 11 . . 3 UnifOn UnifOn
8 simpr 468 . . . . . . . 8
98unieqd 4200 . . . . . . 7
109dmeqd 5042 . . . . . 6
11 simpl 464 . . . . . . . 8
1211unieqd 4200 . . . . . . 7
1312dmeqd 5042 . . . . . 6
1410, 13oveq12d 6326 . . . . 5
1513raleqdv 2979 . . . . . . . 8
1613, 15raleqbidv 2987 . . . . . . 7
1711, 16rexeqbidv 2988 . . . . . 6
188, 17raleqbidv 2987 . . . . 5
1914, 18rabeqbidv 3026 . . . 4
20 df-ucn 21369 . . . 4 Cnu UnifOn UnifOn
2119, 20ovmpt2ga 6445 . . 3 UnifOn UnifOn Cnu
222, 4, 7, 21syl3anc 1292 . 2 UnifOn UnifOn Cnu
23 ustbas2 21318 . . . 4 UnifOn
24 ustbas2 21318 . . . 4 UnifOn
2523, 24oveqan12rd 6328 . . 3 UnifOn UnifOn
2624adantr 472 . . . . . 6 UnifOn UnifOn
2726raleqdv 2979 . . . . . 6 UnifOn UnifOn
2826, 27raleqbidv 2987 . . . . 5 UnifOn UnifOn
2928rexbidv 2892 . . . 4 UnifOn UnifOn
3029ralbidv 2829 . . 3 UnifOn UnifOn
3125, 30rabeqbidv 3026 . 2 UnifOn UnifOn
3222, 31eqtr4d 2508 1 UnifOn UnifOn Cnu
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  wral 2756  wrex 2757  crab 2760  cvv 3031  cuni 4190   class class class wbr 4395   cdm 4839   crn 4840  cfv 5589  (class class class)co 6308   cmap 7490  UnifOncust 21292   Cnucucn 21368 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-ust 21293  df-ucn 21369 This theorem is referenced by:  isucn  21371
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