Theorem iducn 22087

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 e. (UnifOn ` X ) -> ( _I |` X ) e. ( U Cnu U ) )

Proof of Theorem iducn

Dummy variables s r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6174	. . 3 |- ( _I |` X ) : X -1-1-onto-> X
2		f1of 6137	. . 3 |- ( ( _I |` X ) : X -1-1-onto-> X -> ( _I |` X ) : X --> X )
3	1, 2	mp1i 13	. 2 |- ( U e. (UnifOn ` X ) -> ( _I |` X ) : X --> X )
4		simpr 477	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ s e. U ) -> s e. U )
5		fvresi 6439	. . . . . . . 8 |- ( x e. X -> ( ( _I |` X ) ` x ) = x )
6		fvresi 6439	. . . . . . . 8 |- ( y e. X -> ( ( _I |` X ) ` y ) = y )
7	5, 6	breqan12d 4669	. . . . . . 7 |- ( ( x e. X /\ y e. X ) -> ( ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) <-> x s y ) )
8	7	biimprd 238	. . . . . 6 |- ( ( x e. X /\ y e. X ) -> ( x s y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) )
9	8	adantl 482	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ s e. U ) /\ ( x e. X /\ y e. X ) ) -> ( x s y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) )
10	9	ralrimivva 2971	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ s e. U ) -> A. x e. X A. y e. X ( x s y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) )
11		breq 4655	. . . . . . 7 |- ( r = s -> ( x r y <-> x s y ) )
12	11	imbi1d 331	. . . . . 6 |- ( r = s -> ( ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) <-> ( x s y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) ) )
13	12	2ralbidv 2989	. . . . 5 |- ( r = s -> ( A. x e. X A. y e. X ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) <-> A. x e. X A. y e. X ( x s y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) ) )
14	13	rspcev 3309	. . . 4 |- ( ( s e. U /\ A. x e. X A. y e. X ( x s y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) ) -> E. r e. U A. x e. X A. y e. X ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) )
15	4, 10, 14	syl2anc 693	. . 3 |- ( ( U e. (UnifOn ` X ) /\ s e. U ) -> E. r e. U A. x e. X A. y e. X ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) )
16	15	ralrimiva 2966	. 2 |- ( U e. (UnifOn ` X ) -> A. s e. U E. r e. U A. x e. X A. y e. X ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) )
17		isucn 22082	. . 3 |- ( ( U e. (UnifOn ` X ) /\ U e. (UnifOn ` X ) ) -> ( ( _I |` X ) e. ( U Cnu U ) <-> ( ( _I |` X ) : X --> X /\ A. s e. U E. r e. U A. x e. X A. y e. X ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) ) ) )
18	17	anidms 677	. 2 |- ( U e. (UnifOn ` X ) -> ( ( _I |` X ) e. ( U Cnu U ) <-> ( ( _I |` X ) : X --> X /\ A. s e. U E. r e. U A. x e. X A. y e. X ( x r y -> ( ( _I |` X ) ` x ) s ( ( _I |` X ) ` y ) ) ) ) )
19	3, 16, 18	mpbir2and 957	1 |- ( U e. (UnifOn ` X ) -> ( _I |` X ) e. ( U Cnu U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   _I cid 5023   |` cres 5116   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650  UnifOncust 22003   Cn_ucucn 22079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ust 22004  df-ucn 22080

This theorem is referenced by: (None)