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Theorem iducn 24308
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 6887 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
2 f1of 6849 . . 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 7193 . . . . . . . 8 (𝑥𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6 fvresi 7193 . . . . . . . 8 (𝑦𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
75, 6breqan12d 5164 . . . . . . 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 3200 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11 breq 5150 . . . . . . 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 3219 . . . . 5 (𝑟 = 𝑠 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
1413rspcev 3622 . . . 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 3144 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17 isucn 24303 . . 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 2106  wral 3059  wrex 3068   class class class wbr 5148   I cid 5582  cres 5691  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  UnifOncust 24224   Cnucucn 24300
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ust 24225  df-ucn 24301
This theorem is referenced by: (None)
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