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Theorem ucnval 24212
Description: The set of all uniformly continuous function from uniform space π‘ˆ to uniform space 𝑉. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ucnval ((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‘¦))})
Distinct variable groups:   𝑓,π‘Ÿ,𝑠,π‘₯,𝑦,π‘ˆ   𝑓,𝑉,π‘Ÿ,𝑠,π‘₯   𝑓,𝑋,π‘Ÿ,𝑠,π‘₯,𝑦   𝑓,π‘Œ,π‘Ÿ,𝑠,π‘₯
Allowed substitution hints:   𝑉(𝑦)   π‘Œ(𝑦)
Proof of Theorem ucnval
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvunirn 6926 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
21adantr 479 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
3 elfvunirn 6926 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
43adantl 480 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
5 ovex 7450 . . . . 5 (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∈ V
65rabex 5334 . . . 4 {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))} ∈ V
76a1i 11 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))} ∈ V)
8 simpr 483 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑣 = 𝑉)
98unieqd 4921 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑣 = βˆͺ 𝑉)
109dmeqd 5907 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑣 = dom βˆͺ 𝑉)
11 simpl 481 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑒 = π‘ˆ)
1211unieqd 4921 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑒 = βˆͺ π‘ˆ)
1312dmeqd 5907 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑒 = dom βˆͺ π‘ˆ)
1410, 13oveq12d 7435 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (dom βˆͺ 𝑣 ↑m dom βˆͺ 𝑒) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
1513raleqdv 3315 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (βˆ€π‘¦ ∈ dom βˆͺ 𝑒(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
1613, 15raleqbidv 3330 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (βˆ€π‘₯ ∈ dom βˆͺ π‘’βˆ€π‘¦ ∈ dom βˆͺ 𝑒(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
1711, 16rexeqbidv 3331 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (βˆƒπ‘Ÿ ∈ 𝑒 βˆ€π‘₯ ∈ dom βˆͺ π‘’βˆ€π‘¦ ∈ dom βˆͺ 𝑒(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
188, 17raleqbidv 3330 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (βˆ€π‘  ∈ 𝑣 βˆƒπ‘Ÿ ∈ 𝑒 βˆ€π‘₯ ∈ dom βˆͺ π‘’βˆ€π‘¦ ∈ dom βˆͺ 𝑒(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
1914, 18rabeqbidv 3437 . . . 4 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ {𝑓 ∈ (dom βˆͺ 𝑣 ↑m dom βˆͺ 𝑒) ∣ βˆ€π‘  ∈ 𝑣 βˆƒπ‘Ÿ ∈ 𝑒 βˆ€π‘₯ ∈ dom βˆͺ π‘’βˆ€π‘¦ ∈ dom βˆͺ 𝑒(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))} = {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))})
20 df-ucn 24211 . . . 4 Cnu = (𝑒 ∈ βˆͺ ran UnifOn, 𝑣 ∈ βˆͺ ran UnifOn ↦ {𝑓 ∈ (dom βˆͺ 𝑣 ↑m dom βˆͺ 𝑒) ∣ βˆ€π‘  ∈ 𝑣 βˆƒπ‘Ÿ ∈ 𝑒 βˆ€π‘₯ ∈ dom βˆͺ π‘’βˆ€π‘¦ ∈ dom βˆͺ 𝑒(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))})
2119, 20ovmpoga 7573 . . 3 ((π‘ˆ ∈ βˆͺ ran UnifOn ∧ 𝑉 ∈ βˆͺ ran UnifOn ∧ {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))} ∈ V) β†’ (π‘ˆ Cnu𝑉) = {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))})
222, 4, 7, 21syl3anc 1368 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (π‘ˆ Cnu𝑉) = {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))})
23 ustbas2 24160 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ π‘Œ = dom βˆͺ 𝑉)
24 ustbas2 24160 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2523, 24oveqan12rd 7437 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (π‘Œ ↑m 𝑋) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
2624adantr 479 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2726raleqdv 3315 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
2826, 27raleqbidv 3330 . . . . 5 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
2928rexbidv 3169 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) 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‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
3029ralbidv 3168 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) 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‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))))
3125, 30rabeqbidv 3437 . 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‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))} = {𝑓 ∈ (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∣ βˆ€π‘  ∈ 𝑉 βˆƒπ‘Ÿ ∈ π‘ˆ βˆ€π‘₯ ∈ dom βˆͺ π‘ˆβˆ€π‘¦ ∈ dom βˆͺ π‘ˆ(π‘₯π‘Ÿπ‘¦ β†’ (π‘“β€˜π‘₯)𝑠(π‘“β€˜π‘¦))})
3222, 31eqtr4d 2768 1 ((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‘¦))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419  Vcvv 3463  βˆͺ cuni 4908   class class class wbr 5148  dom cdm 5677  ran crn 5678  β€˜cfv 6547  (class class class)co 7417   ↑m cmap 8843  UnifOncust 24134   Cnucucn 24210
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-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-ust 24135  df-ucn 24211
This theorem is referenced by:  isucn  24213
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