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Theorem ucnval 24255

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		elfvunirn 6866	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2	1	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ∈ ∪ ran UnifOn)
3		elfvunirn 6866	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ∈ ∪ ran UnifOn)
4	3	adantl 481	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ∈ ∪ ran UnifOn)
5		ovex 7395	. . . . 5 ⊢ (dom ∪ 𝑉 ↑_m dom ∪ 𝑈) ∈ V
6	5	rabex 5277	. . . 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 4864	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑣 = ∪ 𝑉)
10	9	dmeqd 5856	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑣 = dom ∪ 𝑉)
11		simpl 482	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑢 = 𝑈)
12	11	unieqd 4864	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑢 = ∪ 𝑈)
13	12	dmeqd 5856	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑢 = dom ∪ 𝑈)
14	10, 13	oveq12d 7380	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (dom ∪ 𝑣 ↑_m dom ∪ 𝑢) = (dom ∪ 𝑉 ↑_m dom ∪ 𝑈))
15	13	raleqdv 3296	. . . . . . . 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 3408	. . . 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 24254	. . . 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 7516	. . 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 1374	. 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 24204	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom ∪ 𝑉)
24		ustbas2 24204	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
25	23, 24	oveqan12rd 7382	. . 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 3296	. . . . . 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 3162	. . . 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 3161	. . 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 3408	. 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 2775	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 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ∪ cuni 4851 class class class wbr 5086 dom cdm 5626 ran crn 5627 ‘cfv 6494 (class class class)co 7362 ↑_m cmap 8768 UnifOncust 24179 Cn_ucucn 24253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-iota 6450 df-fun 6496 df-fv 6502 df-ov 7365 df-oprab 7366 df-mpo 7367 df-ust 24180 df-ucn 24254

This theorem is referenced by: isucn 24256

W3C validator