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Theorem ucnval 22489
Description: The set of all uniformly continuous function from uniform space 𝑈 to uniform space 𝑉. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ucnval ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
Distinct variable groups:   𝑓,𝑟,𝑠,𝑥,𝑦,𝑈   𝑓,𝑉,𝑟,𝑠,𝑥   𝑓,𝑋,𝑟,𝑠,𝑥,𝑦   𝑓,𝑌,𝑟,𝑠,𝑥
Allowed substitution hints:   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem ucnval
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrnust 22436 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
21adantr 474 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ran UnifOn)
3 elrnust 22436 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ran UnifOn)
43adantl 475 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ran UnifOn)
5 ovex 6954 . . . . 5 (dom 𝑉𝑚 dom 𝑈) ∈ V
65rabex 5049 . . . 4 {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} ∈ V
76a1i 11 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} ∈ V)
8 simpr 479 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
98unieqd 4681 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
109dmeqd 5571 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑣 = dom 𝑉)
11 simpl 476 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1211unieqd 4681 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1312dmeqd 5571 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑢 = dom 𝑈)
1410, 13oveq12d 6940 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (dom 𝑣𝑚 dom 𝑢) = (dom 𝑉𝑚 dom 𝑈))
1513raleqdv 3340 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1613, 15raleqbidv 3326 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1711, 16rexeqbidv 3327 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → (∃𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
188, 17raleqbidv 3326 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1914, 18rabeqbidv 3392 . . . 4 ((𝑢 = 𝑈𝑣 = 𝑉) → {𝑓 ∈ (dom 𝑣𝑚 dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} = {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
20 df-ucn 22488 . . . 4 Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣𝑚 dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
2119, 20ovmpt2ga 7067 . . 3 ((𝑈 ran UnifOn ∧ 𝑉 ran UnifOn ∧ {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} ∈ V) → (𝑈 Cnu𝑉) = {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
222, 4, 7, 21syl3anc 1439 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
23 ustbas2 22437 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom 𝑉)
24 ustbas2 22437 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
2523, 24oveqan12rd 6942 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌𝑚 𝑋) = (dom 𝑉𝑚 dom 𝑈))
2624adantr 474 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom 𝑈)
2726raleqdv 3340 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2826, 27raleqbidv 3326 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2928rexbidv 3237 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3029ralbidv 3168 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3125, 30rabeqbidv 3392 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))} = {𝑓 ∈ (dom 𝑉𝑚 dom 𝑈) ∣ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
3222, 31eqtr4d 2817 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  {crab 3094  Vcvv 3398   cuni 4671   class class class wbr 4886  dom cdm 5355  ran crn 5356  cfv 6135  (class class class)co 6922  𝑚 cmap 8140  UnifOncust 22411   Cnucucn 22487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-ust 22412  df-ucn 22488
This theorem is referenced by:  isucn  22490
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