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Theorem ucnval 22301
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 22248 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
21adantr 466 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ran UnifOn)
3 elrnust 22248 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ran UnifOn)
43adantl 467 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ran UnifOn)
5 ovex 6827 . . . . 5 (dom 𝑉𝑚 dom 𝑈) ∈ V
65rabex 4947 . . . 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 471 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
98unieqd 4585 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑣 = 𝑉)
109dmeqd 5463 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑣 = dom 𝑉)
11 simpl 468 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1211unieqd 4585 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → 𝑢 = 𝑈)
1312dmeqd 5463 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → dom 𝑢 = dom 𝑈)
1410, 13oveq12d 6814 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (dom 𝑣𝑚 dom 𝑢) = (dom 𝑉𝑚 dom 𝑈))
1513raleqdv 3293 . . . . . . . 8 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1613, 15raleqbidv 3301 . . . . . . 7 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1711, 16rexeqbidv 3302 . . . . . 6 ((𝑢 = 𝑈𝑣 = 𝑉) → (∃𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
188, 17raleqbidv 3301 . . . . 5 ((𝑢 = 𝑈𝑣 = 𝑉) → (∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
1914, 18rabeqbidv 3345 . . . 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 22300 . . . 4 Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣𝑚 dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
2119, 20ovmpt2ga 6941 . . 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 1476 . 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 22249 . . . 4 (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom 𝑉)
24 ustbas2 22249 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
2523, 24oveqan12rd 6816 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌𝑚 𝑋) = (dom 𝑉𝑚 dom 𝑈))
2624adantr 466 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom 𝑈)
2726raleqdv 3293 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2826, 27raleqbidv 3301 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
2928rexbidv 3200 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∃𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3029ralbidv 3135 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦)) ↔ ∀𝑠𝑉𝑟𝑈𝑥 ∈ dom 𝑈𝑦 ∈ dom 𝑈(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))))
3125, 30rabeqbidv 3345 . 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 2808 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 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  Vcvv 3351   cuni 4575   class class class wbr 4787  dom cdm 5250  ran crn 5251  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  UnifOncust 22223   Cnucucn 22299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-ust 22224  df-ucn 22300
This theorem is referenced by:  isucn  22302
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