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Theorem isucn 24103
Description: The predicate "𝐹 is a uniformly continuous function from uniform space π‘ˆ to uniform space 𝑉". (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
isucn ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘ˆ Cnu𝑉) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦)))))
Distinct variable groups:   𝑠,π‘Ÿ,π‘₯,𝑦,𝐹   π‘ˆ,π‘Ÿ,𝑠,π‘₯,𝑦   𝑉,π‘Ÿ,𝑠,π‘₯   𝑋,π‘Ÿ,𝑠,π‘₯,𝑦   π‘Œ,π‘Ÿ,𝑠,π‘₯
Allowed substitution hints:   𝑉(𝑦)   π‘Œ(𝑦)

Proof of Theorem isucn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ucnval 24102 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (π‘ˆ Cnu𝑉) = {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))})
21eleq2d 2818 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘ˆ Cnu𝑉) ↔ 𝐹 ∈ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))}))
3 fveq1 6890 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
4 fveq1 6890 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
53, 4breq12d 5161 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦) ↔ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦)))
65imbi2d 340 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦))))
76ralbidv 3176 . . . . . 6 (𝑓 = 𝐹 β†’ (βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦))))
87rexralbidv 3219 . . . . 5 (𝑓 = 𝐹 β†’ (βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦))))
98ralbidv 3176 . . . 4 (𝑓 = 𝐹 β†’ (βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦))))
109elrab 3683 . . 3 (𝐹 ∈ {𝑓 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))} ↔ (𝐹 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦))))
112, 10bitrdi 287 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘ˆ Cnu𝑉) ↔ (𝐹 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦)))))
12 elfvex 6929 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ π‘Œ ∈ V)
13 elfvex 6929 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 ∈ V)
14 elmapg 8839 . . . 4 ((π‘Œ ∈ V ∧ 𝑋 ∈ V) β†’ (𝐹 ∈ (π‘Œ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
1512, 13, 14syl2anr 596 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘Œ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆπ‘Œ))
1615anbi1d 629 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ ((𝐹 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦))) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦)))))
1711, 16bitrd 279 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘ˆ Cnu𝑉) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (πΉβ€˜π‘₯)𝑠(πΉβ€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431  Vcvv 3473   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ↑m cmap 8826  UnifOncust 24024   Cnucucn 24100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-ust 24025  df-ucn 24101
This theorem is referenced by:  isucn2  24104  ucnima  24106  iducn  24108  cstucnd  24109  ucncn  24110  fmucnd  24117  ucnextcn  24129
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