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Theorem iducn 22595
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 6481 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
2 f1of 6444 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
31, 2mp1i 13 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋):𝑋𝑋)
4 simpr 477 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → 𝑠𝑈)
5 fvresi 6758 . . . . . . . 8 (𝑥𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6 fvresi 6758 . . . . . . . 8 (𝑦𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
75, 6breqan12d 4945 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → ((( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦) ↔ 𝑥𝑠𝑦))
87biimprd 240 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
98adantl 474 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
109ralrimivva 3142 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11 breq 4931 . . . . . . 7 (𝑟 = 𝑠 → (𝑥𝑟𝑦𝑥𝑠𝑦))
1211imbi1d 334 . . . . . 6 (𝑟 = 𝑠 → ((𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
13122ralbidv 3150 . . . . 5 (𝑟 = 𝑠 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
1413rspcev 3536 . . . 4 ((𝑠𝑈 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
154, 10, 14syl2anc 576 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
1615ralrimiva 3133 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17 isucn 22590 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
1817anidms 559 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
193, 16, 18mpbir2and 700 1 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wcel 2050  wral 3089  wrex 3090   class class class wbr 4929   I cid 5311  cres 5409  wf 6184  1-1-ontowf1o 6187  cfv 6188  (class class class)co 6976  UnifOncust 22511   Cnucucn 22587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-map 8208  df-ust 22512  df-ucn 22588
This theorem is referenced by: (None)
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