Theorem iducn 21897

Description: The identity is uniformly continuous from a uniform structure to itself. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)

Assertion

Ref	Expression
iducn	⊢ (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))

Proof of Theorem iducn

Dummy variables 𝑠 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6086	. . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
2		f1of 6050	. . 3 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋⟶𝑋)
3	1, 2	mp1i 13	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋):𝑋⟶𝑋)
4		simpr 476	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝑈)
5		fvresi 6344	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6		fvresi 6344	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
7	5, 6	breqan12d 4599	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦) ↔ 𝑥𝑠𝑦))
8	7	biimprd 237	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
9	8	adantl 481	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
10	9	ralrimivva 2954	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11		breq 4585	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑥𝑟𝑦 ↔ 𝑥𝑠𝑦))
12	11	imbi1d 330	. . . . . 6 ⊢ (𝑟 = 𝑠 → ((𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
13	12	2ralbidv 2972	. . . . 5 ⊢ (𝑟 = 𝑠 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
14	13	rspcev 3282	. . . 4 ⊢ ((𝑠 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
15	4, 10, 14	syl2anc 691	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
16	15	ralrimiva 2949	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠 ∈ 𝑈 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17		isucn 21892	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑠 ∈ 𝑈 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
18	17	anidms 675	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑠 ∈ 𝑈 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
19	3, 16, 18	mpbir2and 959	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   class class class wbr 4583   I cid 4948   ↾ cres 5040  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  UnifOncust 21813   Cn_ucucn 21889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ust 21814  df-ucn 21890

This theorem is referenced by: (None)