ucnval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ucnval		Structured version Visualization version GIF version

Theorem ucnval 24222

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 6863	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
2	1	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑈 ∈ ∪ ran UnifOn)
3		elfvunirn 6863	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑉 ∈ ∪ ran UnifOn)
4	3	adantl 481	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → 𝑉 ∈ ∪ ran UnifOn)
5		ovex 7391	. . . . 5 ⊢ (dom ∪ 𝑉 ↑_m dom ∪ 𝑈) ∈ V
6	5	rabex 5283	. . . 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 4875	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑣 = ∪ 𝑉)
10	9	dmeqd 5853	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑣 = dom ∪ 𝑉)
11		simpl 482	. . . . . . . 8 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → 𝑢 = 𝑈)
12	11	unieqd 4875	. . . . . . 7 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → ∪ 𝑢 = ∪ 𝑈)
13	12	dmeqd 5853	. . . . . 6 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → dom ∪ 𝑢 = dom ∪ 𝑈)
14	10, 13	oveq12d 7376	. . . . 5 ⊢ ((𝑢 = 𝑈 ∧ 𝑣 = 𝑉) → (dom ∪ 𝑣 ↑_m dom ∪ 𝑢) = (dom ∪ 𝑉 ↑_m dom ∪ 𝑈))
15	13	raleqdv 3295	. . . . . . . 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 3315	. . . . . . 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 3316	. . . . . 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 3315	. . . . 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 3416	. . . 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 24221	. . . 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 7512	. . 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 24171	. . . 4 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → 𝑌 = dom ∪ 𝑉)
24		ustbas2 24171	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
25	23, 24	oveqan12rd 7378	. . 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 3295	. . . . . 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 3315	. . . . 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 3159	. . . 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 3158	. . 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 3416	. 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 2773	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 3050 ∃wrex 3059 {crab 3398 Vcvv 3439 ∪ cuni 4862 class class class wbr 5097 dom cdm 5623 ran crn 5624 ‘cfv 6491 (class class class)co 7358 ↑_m cmap 8765 UnifOncust 24146 Cn_ucucn 24220

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 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3399 df-v 3441 df-sbc 3740 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-iota 6447 df-fun 6493 df-fv 6499 df-ov 7361 df-oprab 7362 df-mpo 7363 df-ust 24147 df-ucn 24221

This theorem is referenced by: isucn 24223

W3C validator