Theorem isucn 23775

Description: The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉". (Contributed by Thierry Arnoux, 16-Nov-2017.)

Assertion

Ref	Expression
isucn	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))

Distinct variable groups:   𝑠,𝑟,𝑥,𝑦,𝐹   𝑈,𝑟,𝑠,𝑥,𝑦   𝑉,𝑟,𝑠,𝑥   𝑋,𝑟,𝑠,𝑥,𝑦   𝑌,𝑟,𝑠,𝑥

Allowed substitution hints:   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem isucn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ucnval 23774	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
2	1	eleq2d 2820	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ 𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))}))
3		fveq1 6888	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
4		fveq1 6888	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
5	3, 4	breq12d 5161	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)𝑠(𝑓‘𝑦) ↔ (𝐹‘𝑥)𝑠(𝐹‘𝑦)))
6	5	imbi2d 341	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
7	6	ralbidv 3178	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
8	7	rexralbidv 3221	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
9	8	ralbidv 3178	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
10	9	elrab 3683	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))} ↔ (𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))))
11	2, 10	bitrdi 287	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
12		elfvex 6927	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 ∈ V)
13		elfvex 6927	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14		elmapg 8830	. . . 4 ⊢ ((𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
15	12, 13, 14	syl2anr 598	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑌 ↑m 𝑋) ↔ 𝐹:𝑋⟶𝑌))
16	15	anbi1d 631	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → ((𝐹 ∈ (𝑌 ↑m 𝑋) ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
17	11, 16	bitrd 279	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   class class class wbr 5148  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406   ↑_m cmap 8817  UnifOncust 23696   Cn_ucucn 23772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-map 8819  df-ust 23697  df-ucn 23773

This theorem is referenced by:  isucn2  23776  ucnima  23778  iducn  23780  cstucnd  23781  ucncn  23782  fmucnd  23789  ucnextcn  23801