MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isucn Structured version   Visualization version   GIF version

Theorem isucn 24214
Description: The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉". (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
isucn ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
Distinct variable groups:   𝑠,𝑟,𝑥,𝑦,𝐹   𝑈,𝑟,𝑠,𝑥,𝑦   𝑉,𝑟,𝑠,𝑥   𝑋,𝑟,𝑠,𝑥,𝑦   𝑌,𝑟,𝑠,𝑥
Allowed substitution hints:   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem isucn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ucnval 24213 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
21eleq2d 2811 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))}))
3 fveq1 6893 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
4 fveq1 6893 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
53, 4breq12d 5161 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝑠(𝑓𝑦) ↔ (𝐹𝑥)𝑠(𝐹𝑦)))
65imbi2d 339 . . . . . . 7 (𝑓 = 𝐹 → ((𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
76ralbidv 3168 . . . . . 6 (𝑓 = 𝐹 → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
87rexralbidv 3211 . . . . 5 (𝑓 = 𝐹 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
98ralbidv 3168 . . . 4 (𝑓 = 𝐹 → (∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
109elrab 3680 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌m 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
112, 10bitrdi 286 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
12 elfvex 6932 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 ∈ V)
13 elfvex 6932 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14 elmapg 8856 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
1512, 13, 14syl2anr 595 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑌m 𝑋) ↔ 𝐹:𝑋𝑌))
1615anbi1d 629 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → ((𝐹 ∈ (𝑌m 𝑋) ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
1711, 16bitrd 278 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3051  wrex 3060  {crab 3419  Vcvv 3463   class class class wbr 5148  wf 6543  cfv 6547  (class class class)co 7417  m cmap 8843  UnifOncust 24135   Cnucucn 24211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8845  df-ust 24136  df-ucn 24212
This theorem is referenced by:  isucn2  24215  ucnima  24217  iducn  24219  cstucnd  24220  ucncn  24221  fmucnd  24228  ucnextcn  24240
  Copyright terms: Public domain W3C validator
OSZAR »