Theorem iducn 22886

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 6646	. . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
2		f1of 6609	. . 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 487	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) → 𝑠 ∈ 𝑈)
5		fvresi 6929	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6		fvresi 6929	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
7	5, 6	breqan12d 5074	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦) ↔ 𝑥𝑠𝑦))
8	7	biimprd 250	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
9	8	adantl 484	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
10	9	ralrimivva 3191	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11		breq 5060	. . . . . . 7 ⊢ (𝑟 = 𝑠 → (𝑥𝑟𝑦 ↔ 𝑥𝑠𝑦))
12	11	imbi1d 344	. . . . . 6 ⊢ (𝑟 = 𝑠 → ((𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
13	12	2ralbidv 3199	. . . . 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 3622	. . . 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 586	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠 ∈ 𝑈) → ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
16	15	ralrimiva 3182	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠 ∈ 𝑈 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17		isucn 22881	. . 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 569	. 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 711	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   class class class wbr 5058   I cid 5453   ↾ cres 5551  ⟶wf 6345  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150  UnifOncust 22802   Cn_ucucn 22878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-ust 22803  df-ucn 22879

This theorem is referenced by: (None)