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Theorem ucnval 24200
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 6932 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
21adantr 479 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ran UnifOn)
3 elfvunirn 6932 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ran UnifOn)
43adantl 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ran UnifOn)
5 ovex 7457 . . . . 5 (dom 𝑉m dom 𝑈) ∈ V
65rabex 5336 . . . 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 483 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
98unieqd 4923 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
109dmeqd 5910 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑣 = dom 𝑉)
11 simpl 481 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1211unieqd 4923 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1312dmeqd 5910 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑢 = dom 𝑈)
1410, 13oveq12d 7442 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (dom 𝑣m dom 𝑢) = (dom 𝑉m dom 𝑈))
1513raleqdv 3321 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1613, 15raleqbidv 3338 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1711, 16rexeqbidv 3339 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → (∃𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
188, 17raleqbidv 3338 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1914, 18rabeqbidv 3446 . . . 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 24199 . . . 4 Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣m dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
2119, 20ovmpoga 7579 . . 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 1368 . 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 24148 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom 𝑉)
24 ustbas2 24148 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
2523, 24oveqan12rd 7444 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌m 𝑋) = (dom 𝑉m dom 𝑈))
2624adantr 479 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom 𝑈)
2726raleqdv 3321 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2826, 27raleqbidv 3338 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2928rexbidv 3174 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3029ralbidv 3173 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3125, 30rabeqbidv 3446 . 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 2770 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 394   = wceq 1533  wcel 2098  wral 3057  wrex 3066  {crab 3428  Vcvv 3471   cuni 4910   class class class wbr 5150  dom cdm 5680  ran crn 5681  cfv 6551  (class class class)co 7424  m cmap 8849  UnifOncust 24122   Cnucucn 24198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-ust 24123  df-ucn 24199
This theorem is referenced by:  isucn  24201
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