Theorem ucnval 24622

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 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})

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.

Step	Hyp	Ref	Expression
1		elfvunirn 6935	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2	1	adantr 486	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ∈ ∪ ran UnifOn)
3		elfvunirn 6935	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ∈ ∪ ran UnifOn)
4	3	adantl 487	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ∈ ∪ ran UnifOn)
5		ovex 7473	. . . . 5 ⊢ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∈ V
6	5	rabex 5304	. . . 4 ⊢ {𝑓 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))} ∈ V
7	6	a1i 11	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → {𝑓 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))} ∈ V)
8		simpr 490	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
9	8	unieqd 4883	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑣 = ∪ 𝑉)
10	9	dmeqd 5893	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑣 = dom ∪ 𝑉)
11		simpl 488	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑢 = 𝑈)
12	11	unieqd 4883	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑢 = ∪ 𝑈)
13	12	dmeqd 5893	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑢 = dom ∪ 𝑈)
14	10, 13	oveq12d 7458	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (dom ∪ 𝑣 ↑m dom ∪ 𝑢) = (dom ∪ 𝑉 ↑m dom ∪ 𝑈))
15	13	raleqdv 3323	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
16	13, 15	raleqbidv 3338	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
17	11, 16	rexeqbidv 3339	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
18	8, 17	raleqbidv 3338	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
19	14, 18	rabeqbidv 3433	. . . 4 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → {𝑓 ∈ (dom ∪ 𝑣 ↑m dom ∪ 𝑢) ∣ ∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))} = {𝑓 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
20		df-ucn 24621	. . . 4 ⊢ Cnu = (𝑢 ∈ ∪ ran UnifOn, 𝑣 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (dom ∪ 𝑣 ↑m dom ∪ 𝑢) ∣ ∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
21	19, 20	ovmpoga 7595	. . 3 ⊢ ((𝑈 ∈ ∪ ran UnifOn ∧ 𝑉 ∈ ∪ ran UnifOn ∧ {𝑓 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))} ∈ V) → (𝑈 Cnu𝑉) = {𝑓 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
22	2, 4, 7, 21	syl3anc 1399	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
23		ustbas2 24571	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom ∪ 𝑉)
24		ustbas2 24571	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
25	23, 24	oveqan12rd 7460	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌 ↑m 𝑋) = (dom ∪ 𝑉 ↑m dom ∪ 𝑈))
26	24	adantr 486	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom ∪ 𝑈)
27	26	raleqdv 3323	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
28	26, 27	raleqbidv 3338	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
29	28	rexbidv 3190	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
30	29	ralbidv 3189	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦)) ↔ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))))
31	25, 30	rabeqbidv 3433	. 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 ∈ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
32	22, 31	eqtr4d 2801	1 ⊢ ((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))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  ∪ cuni 4870   class class class wbr 5106  dom cdm 5655  ran crn 5656  ‘cfv 6555  (class class class)co 7440   ↑_m cmap 8872  UnifOncust 24546   Cn_ucucn 24620

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6509  df-fun 6557  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-ust 24547  df-ucn 24621

This theorem is used by:  isucn  24623