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

Theorem iducn 24293
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 6885 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
2 f1of 6847 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
31, 2mp1i 13 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋):𝑋𝑋)
4 simpr 484 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → 𝑠𝑈)
5 fvresi 7194 . . . . . . . 8 (𝑥𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6 fvresi 7194 . . . . . . . 8 (𝑦𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
75, 6breqan12d 5158 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → ((( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦) ↔ 𝑥𝑠𝑦))
87biimprd 248 . . . . . 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 3201 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11 breq 5144 . . . . . . 7 (𝑟 = 𝑠 → (𝑥𝑟𝑦𝑥𝑠𝑦))
1211imbi1d 341 . . . . . 6 (𝑟 = 𝑠 → ((𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
13122ralbidv 3220 . . . . 5 (𝑟 = 𝑠 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
1413rspcev 3621 . . . 4 ((𝑠𝑈 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
154, 10, 14syl2anc 584 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
1615ralrimiva 3145 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17 isucn 24288 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
1817anidms 566 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
193, 16, 18mpbir2and 713 1 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2107  wral 3060  wrex 3069   class class class wbr 5142   I cid 5576  cres 5686  wf 6556  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  UnifOncust 24209   Cnucucn 24285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-ust 24210  df-ucn 24286
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator