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Theorem ucnval 23439

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 Cn_u𝑉) = {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		elrnust 23386	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2	1	adantr 481	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ∈ ∪ ran UnifOn)
3		elrnust 23386	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ∈ ∪ ran UnifOn)
4	3	adantl 482	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ∈ ∪ ran UnifOn)
5		ovex 7300	. . . . 5 ⊢ (dom ∪ 𝑉 ↑_m dom ∪ 𝑈) ∈ V
6	5	rabex 5254	. . . 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 485	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑣 = 𝑉)
9	8	unieqd 4853	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑣 = ∪ 𝑉)
10	9	dmeqd 5807	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑣 = dom ∪ 𝑉)
11		simpl 483	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑢 = 𝑈)
12	11	unieqd 4853	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑢 = ∪ 𝑈)
13	12	dmeqd 5807	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑢 = dom ∪ 𝑈)
14	10, 13	oveq12d 7285	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (dom ∪ 𝑣 ↑_m dom ∪ 𝑢) = (dom ∪ 𝑉 ↑_m dom ∪ 𝑈))
15	13	raleqdv 3346	. . . . . . . 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 3334	. . . . . . 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 3335	. . . . . 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 3334	. . . . 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 3417	. . . 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 23438	. . . 4 ⊢ Cn_u = (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 7417	. . 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 Cn_u𝑉) = {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 1370	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cn_u𝑉) = {𝑓 ∈ (dom ∪ 𝑉 ↑_m dom ∪ 𝑈) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ dom ∪ 𝑈∀𝑦 ∈ dom ∪ 𝑈(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
23		ustbas2 23387	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom ∪ 𝑉)
24		ustbas2 23387	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
25	23, 24	oveqan12rd 7287	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑌 ↑_m 𝑋) = (dom ∪ 𝑉 ↑_m dom ∪ 𝑈))
26	24	adantr 481	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑋 = dom ∪ 𝑈)
27	26	raleqdv 3346	. . . . . 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 3334	. . . . 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 3224	. . . 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 3121	. . 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 3417	. 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 2781	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cn_u𝑉) = {𝑓 ∈ (𝑌 ↑_m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3429 ∪ cuni 4839 class class class wbr 5073 dom cdm 5584 ran crn 5585 ‘cfv 6426 (class class class)co 7267 ↑_m cmap 8602 UnifOncust 23361 Cn_ucucn 23437

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-iota 6384 df-fun 6428 df-fn 6429 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-ust 23362 df-ucn 23438

This theorem is referenced by: isucn 23440

W3C validator