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Theorem isucn 22063
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 22062 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
21eleq2d 2685 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))}))
3 fveq1 6177 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
4 fveq1 6177 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
53, 4breq12d 4657 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓𝑥)𝑠(𝑓𝑦) ↔ (𝐹𝑥)𝑠(𝐹𝑦)))
65imbi2d 330 . . . . . . 7 (𝑓 = 𝐹 → ((𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
76ralbidv 2983 . . . . . 6 (𝑓 = 𝐹 → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
87rexralbidv 3054 . . . . 5 (𝑓 = 𝐹 → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
98ralbidv 2983 . . . 4 (𝑓 = 𝐹 → (∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
109elrab 3357 . . 3 (𝐹 ∈ {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} ↔ (𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
112, 10syl6bb 276 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
12 elfvex 6208 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 ∈ V)
13 elfvex 6208 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14 elmapg 7855 . . . 4 ((𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
1512, 13, 14syl2anr 495 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑌𝑚 𝑋) ↔ 𝐹:𝑋𝑌))
1615anbi1d 740 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → ((𝐹 ∈ (𝑌𝑚 𝑋) ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
1711, 16bitrd 268 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 196  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  {crab 2913  Vcvv 3195   class class class wbr 4644  wf 5872  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  UnifOncust 21984   Cnucucn 22060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-ust 21985  df-ucn 22061
This theorem is referenced by:  isucn2  22064  ucnima  22066  iducn  22068  cstucnd  22069  ucncn  22070  fmucnd  22077  ucnextcn  22089
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