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Theorem ucnval 23773
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 6920 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
21adantr 481 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
3 elfvunirn 6920 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
43adantl 482 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
5 ovex 7438 . . . . 5 (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∈ V
65rabex 5331 . . . 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 485 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑣 = 𝑉)
98unieqd 4921 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑣 = βˆͺ 𝑉)
109dmeqd 5903 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑣 = dom βˆͺ 𝑉)
11 simpl 483 . . . . . . . 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 5903 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑒 = dom βˆͺ π‘ˆ)
1410, 13oveq12d 7423 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (dom βˆͺ 𝑣 ↑m dom βˆͺ 𝑒) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
1513raleqdv 3325 . . . . . . . 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 3342 . . . . . . 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 3343 . . . . . 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 3342 . . . . 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 3449 . . . 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 23772 . . . 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 7558 . . 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 1371 . 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 23721 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ π‘Œ = dom βˆͺ 𝑉)
24 ustbas2 23721 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2523, 24oveqan12rd 7425 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (π‘Œ ↑m 𝑋) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
2624adantr 481 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2726raleqdv 3325 . . . . . 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 3342 . . . . 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 3178 . . . 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 3177 . . 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 3449 . 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 2775 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474  βˆͺ cuni 4907   class class class wbr 5147  dom cdm 5675  ran crn 5676  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  UnifOncust 23695   Cnucucn 23771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ust 23696  df-ucn 23772
This theorem is referenced by:  isucn  23774
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