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Theorem ucnval 24300
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 elfvunirn 6951 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
21adantr 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ran UnifOn)
3 elfvunirn 6951 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ran UnifOn)
43adantl 481 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ran UnifOn)
5 ovex 7478 . . . . 5 (dom 𝑉m dom 𝑈) ∈ V
65rabex 5360 . . . 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 484 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
98unieqd 4944 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
109dmeqd 5929 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑣 = dom 𝑉)
11 simpl 482 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1211unieqd 4944 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1312dmeqd 5929 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑢 = dom 𝑈)
1410, 13oveq12d 7463 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (dom 𝑣m dom 𝑢) = (dom 𝑉m dom 𝑈))
1513raleqdv 3329 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1613, 15raleqbidv 3349 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1711, 16rexeqbidv 3350 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → (∃𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
188, 17raleqbidv 3349 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1914, 18rabeqbidv 3456 . . . 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 24299 . . . 4 Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣m dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
2119, 20ovmpoga 7600 . . 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 1371 . 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 24248 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom 𝑉)
24 ustbas2 24248 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
2523, 24oveqan12rd 7465 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌m 𝑋) = (dom 𝑉m dom 𝑈))
2624adantr 480 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom 𝑈)
2726raleqdv 3329 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2826, 27raleqbidv 3349 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2928rexbidv 3181 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3029ralbidv 3180 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3125, 30rabeqbidv 3456 . 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 2777 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 395   = wceq 1537  wcel 2103  wral 3063  wrex 3072  {crab 3438  Vcvv 3482   cuni 4931   class class class wbr 5169  dom cdm 5699  ran crn 5700  cfv 6572  (class class class)co 7445  m cmap 8880  UnifOncust 24222   Cnucucn 24298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-ust 24223  df-ucn 24299
This theorem is referenced by:  isucn  24301
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