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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.
StepHypRef Expression
1 f1oi 6086 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
2 f1of 6050 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
31, 2mp1i 13 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋):𝑋𝑋)
4 simpr 476 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → 𝑠𝑈)
5 fvresi 6344 . . . . . . . 8 (𝑥𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6 fvresi 6344 . . . . . . . 8 (𝑦𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
75, 6breqan12d 4599 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → ((( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦) ↔ 𝑥𝑠𝑦))
87biimprd 237 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
98adantl 481 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
109ralrimivva 2954 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11 breq 4585 . . . . . . 7 (𝑟 = 𝑠 → (𝑥𝑟𝑦𝑥𝑠𝑦))
1211imbi1d 330 . . . . . 6 (𝑟 = 𝑠 → ((𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
13122ralbidv 2972 . . . . 5 (𝑟 = 𝑠 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
1413rspcev 3282 . . . 4 ((𝑠𝑈 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
154, 10, 14syl2anc 691 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
1615ralrimiva 2949 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17 isucn 21892 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
1817anidms 675 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
193, 16, 18mpbir2and 959 1 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))
 Colors of variables: wff setvar class 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   Cnucucn 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)
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