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Theorem ucnval 23782
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 6924 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
21adantr 482 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
3 elfvunirn 6924 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
43adantl 483 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
5 ovex 7442 . . . . 5 (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∈ V
65rabex 5333 . . . 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 486 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑣 = 𝑉)
98unieqd 4923 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑣 = βˆͺ 𝑉)
109dmeqd 5906 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑣 = dom βˆͺ 𝑉)
11 simpl 484 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑒 = π‘ˆ)
1211unieqd 4923 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑒 = βˆͺ π‘ˆ)
1312dmeqd 5906 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑒 = dom βˆͺ π‘ˆ)
1410, 13oveq12d 7427 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (dom βˆͺ 𝑣 ↑m dom βˆͺ 𝑒) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
1513raleqdv 3326 . . . . . . . 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 3343 . . . . . . 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 3344 . . . . . 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 3343 . . . . 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 3450 . . . 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 23781 . . . 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 7562 . . 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 1372 . 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 23730 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ π‘Œ = dom βˆͺ 𝑉)
24 ustbas2 23730 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2523, 24oveqan12rd 7429 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (π‘Œ ↑m 𝑋) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
2624adantr 482 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2726raleqdv 3326 . . . . . 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 3343 . . . . 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 3179 . . . 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 3178 . . 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 3450 . 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 2776 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475  βˆͺ cuni 4909   class class class wbr 5149  dom cdm 5677  ran crn 5678  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  UnifOncust 23704   Cnucucn 23780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  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 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-ust 23705  df-ucn 23781
This theorem is referenced by:  isucn  23783
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