Theorem ucnval 24191

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 6852	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2	1	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ∈ ∪ ran UnifOn)
3		elfvunirn 6852	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ∈ ∪ ran UnifOn)
4	3	adantl 481	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ∈ ∪ ran UnifOn)
5		ovex 7379	. . . . 5 ⊢ (dom ∪ 𝑉 ↑m dom ∪ 𝑈) ∈ V
6	5	rabex 5275	. . . 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 484	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
9	8	unieqd 4869	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑣 = ∪ 𝑉)
10	9	dmeqd 5844	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑣 = dom ∪ 𝑉)
11		simpl 482	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑢 = 𝑈)
12	11	unieqd 4869	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑢 = ∪ 𝑈)
13	12	dmeqd 5844	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑢 = dom ∪ 𝑈)
14	10, 13	oveq12d 7364	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (dom ∪ 𝑣 ↑m dom ∪ 𝑢) = (dom ∪ 𝑉 ↑m dom ∪ 𝑈))
15	13	raleqdv 3292	. . . . . . . 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 3312	. . . . . . 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 3313	. . . . . 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 3312	. . . . 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 3413	. . . 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 24190	. . . 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 7500	. . 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 1373	. 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 24140	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom ∪ 𝑉)
24		ustbas2 24140	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
25	23, 24	oveqan12rd 7366	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌 ↑m 𝑋) = (dom ∪ 𝑉 ↑m dom ∪ 𝑈))
26	24	adantr 480	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom ∪ 𝑈)
27	26	raleqdv 3292	. . . . . 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 3312	. . . . 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 3156	. . . 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 3155	. . 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 3413	. 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 2769	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))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436  ∪ cuni 4856   class class class wbr 5089  dom cdm 5614  ran crn 5615  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  UnifOncust 24115   Cn_ucucn 24189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-ust 24116  df-ucn 24190

This theorem is referenced by:  isucn  24192