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Theorem ucnval 24169
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 6923 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
21adantr 480 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ π‘ˆ ∈ βˆͺ ran UnifOn)
3 elfvunirn 6923 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
43adantl 481 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑉 ∈ βˆͺ ran UnifOn)
5 ovex 7447 . . . . 5 (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ) ∈ V
65rabex 5328 . . . 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 484 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑣 = 𝑉)
98unieqd 4916 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑣 = βˆͺ 𝑉)
109dmeqd 5902 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑣 = dom βˆͺ 𝑉)
11 simpl 482 . . . . . . . 8 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ 𝑒 = π‘ˆ)
1211unieqd 4916 . . . . . . 7 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ βˆͺ 𝑒 = βˆͺ π‘ˆ)
1312dmeqd 5902 . . . . . 6 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ dom βˆͺ 𝑒 = dom βˆͺ π‘ˆ)
1410, 13oveq12d 7432 . . . . 5 ((𝑒 = π‘ˆ ∧ 𝑣 = 𝑉) β†’ (dom βˆͺ 𝑣 ↑m dom βˆͺ 𝑒) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
1513raleqdv 3320 . . . . . . . 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 3337 . . . . . . 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 3338 . . . . . 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 3337 . . . . 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 3444 . . . 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 24168 . . . 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 7569 . . 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 1369 . 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 24117 . . . 4 (𝑉 ∈ (UnifOnβ€˜π‘Œ) β†’ π‘Œ = dom βˆͺ 𝑉)
24 ustbas2 24117 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2523, 24oveqan12rd 7434 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ (π‘Œ ↑m 𝑋) = (dom βˆͺ 𝑉 ↑m dom βˆͺ π‘ˆ))
2624adantr 480 . . . . . 6 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ (UnifOnβ€˜π‘Œ)) β†’ 𝑋 = dom βˆͺ π‘ˆ)
2726raleqdv 3320 . . . . . 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 3337 . . . . 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 3173 . . . 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 3172 . . 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 3444 . 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 2770 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 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  βˆƒwrex 3065  {crab 3427  Vcvv 3469  βˆͺ cuni 4903   class class class wbr 5142  dom cdm 5672  ran crn 5673  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836  UnifOncust 24091   Cnucucn 24167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-ust 24092  df-ucn 24168
This theorem is referenced by:  isucn  24170
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