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Theorem iducn 23635
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 6822 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
2 f1of 6784 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
31, 2mp1i 13 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋):𝑋𝑋)
4 simpr 485 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → 𝑠𝑈)
5 fvresi 7119 . . . . . . . 8 (𝑥𝑋 → (( I ↾ 𝑋)‘𝑥) = 𝑥)
6 fvresi 7119 . . . . . . . 8 (𝑦𝑋 → (( I ↾ 𝑋)‘𝑦) = 𝑦)
75, 6breqan12d 5121 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → ((( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦) ↔ 𝑥𝑠𝑦))
87biimprd 247 . . . . . 6 ((𝑥𝑋𝑦𝑋) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
98adantl 482 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
109ralrimivva 3197 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑠𝑈) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
11 breq 5107 . . . . . . 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 3212 . . . . 5 (𝑟 = 𝑠 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑠𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦))))
1413rspcev 3581 . . . 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 3143 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))
17 isucn 23630 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
1817anidms 567 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈) ↔ (( I ↾ 𝑋):𝑋𝑋 ∧ ∀𝑠𝑈𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (( I ↾ 𝑋)‘𝑥)𝑠(( I ↾ 𝑋)‘𝑦)))))
193, 16, 18mpbir2and 711 1 (𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105   I cid 5530  cres 5635  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  UnifOncust 23551   Cnucucn 23627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-ust 23552  df-ucn 23628
This theorem is referenced by: (None)
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