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Theorem fmucnd 20773
Description: The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)
Hypotheses
Ref Expression
fmucnd.1  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
fmucnd.2  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
fmucnd.3  |-  ( ph  ->  F  e.  ( U Cnu V ) )
fmucnd.4  |-  ( ph  ->  C  e.  (CauFilu `  U
) )
fmucnd.5  |-  D  =  ran  ( a  e.  C  |->  ( F "
a ) )
Assertion
Ref Expression
fmucnd  |-  ( ph  ->  D  e.  (CauFilu `  V
) )
Distinct variable groups:    C, a    D, a    F, a    V, a    X, a    Y, a    ph, a
Allowed substitution hint:    U( a)

Proof of Theorem fmucnd
Dummy variables  c 
b  v  r  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmucnd.1 . . . 4  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
2 fmucnd.4 . . . 4  |-  ( ph  ->  C  e.  (CauFilu `  U
) )
3 cfilufbas 20770 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  C  e.  (CauFilu `  U ) )  ->  C  e.  (
fBas `  X )
)
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  C  e.  ( fBas `  X ) )
5 fmucnd.2 . . . 4  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
6 fmucnd.3 . . . 4  |-  ( ph  ->  F  e.  ( U Cnu V ) )
7 isucn 20759 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y )
)  ->  ( F  e.  ( U Cnu V )  <->  ( F : X --> Y  /\  A. v  e.  V  E. r  e.  U  A. x  e.  X  A. y  e.  X  (
x r y  -> 
( F `  x
) v ( F `
 y ) ) ) ) )
87simprbda 623 . . . 4  |-  ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  (UnifOn `  Y )
)  /\  F  e.  ( U Cnu V ) )  ->  F : X --> Y )
91, 5, 6, 8syl21anc 1228 . . 3  |-  ( ph  ->  F : X --> Y )
105elfvexd 5884 . . 3  |-  ( ph  ->  Y  e.  _V )
11 fmucnd.5 . . . 4  |-  D  =  ran  ( a  e.  C  |->  ( F "
a ) )
1211fbasrn 20363 . . 3  |-  ( ( C  e.  ( fBas `  X )  /\  F : X --> Y  /\  Y  e.  _V )  ->  D  e.  ( fBas `  Y
) )
134, 9, 10, 12syl3anc 1229 . 2  |-  ( ph  ->  D  e.  ( fBas `  Y ) )
14 simplr 755 . . . . . . . 8  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  a  e.  C )
15 eqid 2443 . . . . . . . 8  |-  ( F
" a )  =  ( F " a
)
16 imaeq2 5323 . . . . . . . . . 10  |-  ( c  =  a  ->  ( F " c )  =  ( F " a
) )
1716eqeq2d 2457 . . . . . . . . 9  |-  ( c  =  a  ->  (
( F " a
)  =  ( F
" c )  <->  ( F " a )  =  ( F " a ) ) )
1817rspcev 3196 . . . . . . . 8  |-  ( ( a  e.  C  /\  ( F " a )  =  ( F "
a ) )  ->  E. c  e.  C  ( F " a )  =  ( F "
c ) )
1914, 15, 18sylancl 662 . . . . . . 7  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  E. c  e.  C  ( F " a )  =  ( F " c ) )
20 imaexg 6722 . . . . . . . . 9  |-  ( F  e.  ( U Cnu V )  ->  ( F "
a )  e.  _V )
21 eqid 2443 . . . . . . . . . 10  |-  ( c  e.  C  |->  ( F
" c ) )  =  ( c  e.  C  |->  ( F "
c ) )
2221elrnmpt 5239 . . . . . . . . 9  |-  ( ( F " a )  e.  _V  ->  (
( F " a
)  e.  ran  (
c  e.  C  |->  ( F " c ) )  <->  E. c  e.  C  ( F " a )  =  ( F "
c ) ) )
236, 20, 223syl 20 . . . . . . . 8  |-  ( ph  ->  ( ( F "
a )  e.  ran  ( c  e.  C  |->  ( F " c
) )  <->  E. c  e.  C  ( F " a )  =  ( F " c ) ) )
2423ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ( ( F " a )  e. 
ran  ( c  e.  C  |->  ( F "
c ) )  <->  E. c  e.  C  ( F " a )  =  ( F " c ) ) )
2519, 24mpbird 232 . . . . . 6  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ( F " a )  e.  ran  ( c  e.  C  |->  ( F " c
) ) )
26 imaeq2 5323 . . . . . . . . 9  |-  ( a  =  c  ->  ( F " a )  =  ( F " c
) )
2726cbvmptv 4528 . . . . . . . 8  |-  ( a  e.  C  |->  ( F
" a ) )  =  ( c  e.  C  |->  ( F "
c ) )
2827rneqi 5219 . . . . . . 7  |-  ran  (
a  e.  C  |->  ( F " a ) )  =  ran  (
c  e.  C  |->  ( F " c ) )
2911, 28eqtri 2472 . . . . . 6  |-  D  =  ran  ( c  e.  C  |->  ( F "
c ) )
3025, 29syl6eleqr 2542 . . . . 5  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ( F " a )  e.  D
)
31 ffn 5721 . . . . . . . . 9  |-  ( F : X --> Y  ->  F  Fn  X )
329, 31syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  X )
3332ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  F  Fn  X )
34 simplll 759 . . . . . . . 8  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ph )
35 fbelss 20312 . . . . . . . . 9  |-  ( ( C  e.  ( fBas `  X )  /\  a  e.  C )  ->  a  C_  X )
364, 35sylan 471 . . . . . . . 8  |-  ( (
ph  /\  a  e.  C )  ->  a  C_  X )
3734, 14, 36syl2anc 661 . . . . . . 7  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  a  C_  X )
38 fmucndlem 20772 . . . . . . 7  |-  ( ( F  Fn  X  /\  a  C_  X )  -> 
( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " ( a  X.  a ) )  =  ( ( F
" a )  X.  ( F " a
) ) )
3933, 37, 38syl2anc 661 . . . . . 6  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ( (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
) " ( a  X.  a ) )  =  ( ( F
" a )  X.  ( F " a
) ) )
40 eqid 2443 . . . . . . . . 9  |-  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. )
4140mpt2fun 6389 . . . . . . . 8  |-  Fun  (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
42 funimass2 5652 . . . . . . . 8  |-  ( ( Fun  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. )  /\  (
a  X.  a ) 
C_  ( `' ( x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
) " v ) )  ->  ( (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
) " ( a  X.  a ) ) 
C_  v )
4341, 42mpan 670 . . . . . . 7  |-  ( ( a  X.  a ) 
C_  ( `' ( x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
) " v )  ->  ( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. ) " ( a  X.  a ) )  C_  v )
4443adantl 466 . . . . . 6  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ( (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
) " ( a  X.  a ) ) 
C_  v )
4539, 44eqsstr3d 3524 . . . . 5  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  ( ( F " a )  X.  ( F " a
) )  C_  v
)
46 id 22 . . . . . . . 8  |-  ( b  =  ( F "
a )  ->  b  =  ( F "
a ) )
4746sqxpeqd 5015 . . . . . . 7  |-  ( b  =  ( F "
a )  ->  (
b  X.  b )  =  ( ( F
" a )  X.  ( F " a
) ) )
4847sseq1d 3516 . . . . . 6  |-  ( b  =  ( F "
a )  ->  (
( b  X.  b
)  C_  v  <->  ( ( F " a )  X.  ( F " a
) )  C_  v
) )
4948rspcev 3196 . . . . 5  |-  ( ( ( F " a
)  e.  D  /\  ( ( F "
a )  X.  ( F " a ) ) 
C_  v )  ->  E. b  e.  D  ( b  X.  b
)  C_  v )
5030, 45, 49syl2anc 661 . . . 4  |-  ( ( ( ( ph  /\  v  e.  V )  /\  a  e.  C
)  /\  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )  ->  E. b  e.  D  ( b  X.  b )  C_  v
)
511adantr 465 . . . . 5  |-  ( (
ph  /\  v  e.  V )  ->  U  e.  (UnifOn `  X )
)
522adantr 465 . . . . 5  |-  ( (
ph  /\  v  e.  V )  ->  C  e.  (CauFilu `  U ) )
535adantr 465 . . . . . 6  |-  ( (
ph  /\  v  e.  V )  ->  V  e.  (UnifOn `  Y )
)
546adantr 465 . . . . . 6  |-  ( (
ph  /\  v  e.  V )  ->  F  e.  ( U Cnu V ) )
55 simpr 461 . . . . . 6  |-  ( (
ph  /\  v  e.  V )  ->  v  e.  V )
56 nfcv 2605 . . . . . . 7  |-  F/_ s <. ( F `  x
) ,  ( F `
 y ) >.
57 nfcv 2605 . . . . . . 7  |-  F/_ t <. ( F `  x
) ,  ( F `
 y ) >.
58 nfcv 2605 . . . . . . 7  |-  F/_ x <. ( F `  s
) ,  ( F `
 t ) >.
59 nfcv 2605 . . . . . . 7  |-  F/_ y <. ( F `  s
) ,  ( F `
 t ) >.
60 simpl 457 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  t )  ->  x  =  s )
6160fveq2d 5860 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  t )  ->  ( F `  x
)  =  ( F `
 s ) )
62 simpr 461 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  t )  ->  y  =  t )
6362fveq2d 5860 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  t )  ->  ( F `  y
)  =  ( F `
 t ) )
6461, 63opeq12d 4210 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  t )  -> 
<. ( F `  x
) ,  ( F `
 y ) >.  =  <. ( F `  s ) ,  ( F `  t )
>. )
6556, 57, 58, 59, 64cbvmpt2 6361 . . . . . 6  |-  ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( s  e.  X ,  t  e.  X  |->  <. ( F `  s ) ,  ( F `  t )
>. )
6651, 53, 54, 55, 65ucnprima 20763 . . . . 5  |-  ( (
ph  /\  v  e.  V )  ->  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v )  e.  U )
67 cfiluexsm 20771 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  C  e.  (CauFilu `  U )  /\  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v )  e.  U )  ->  E. a  e.  C  ( a  X.  a
)  C_  ( `' ( x  e.  X ,  y  e.  X  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
) " v ) )
6851, 52, 66, 67syl3anc 1229 . . . 4  |-  ( (
ph  /\  v  e.  V )  ->  E. a  e.  C  ( a  X.  a )  C_  ( `' ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " v ) )
6950, 68r19.29a 2985 . . 3  |-  ( (
ph  /\  v  e.  V )  ->  E. b  e.  D  ( b  X.  b )  C_  v
)
7069ralrimiva 2857 . 2  |-  ( ph  ->  A. v  e.  V  E. b  e.  D  ( b  X.  b
)  C_  v )
71 iscfilu 20769 . . 3  |-  ( V  e.  (UnifOn `  Y
)  ->  ( D  e.  (CauFilu `  V )  <->  ( D  e.  ( fBas `  Y
)  /\  A. v  e.  V  E. b  e.  D  ( b  X.  b )  C_  v
) ) )
725, 71syl 16 . 2  |-  ( ph  ->  ( D  e.  (CauFilu `  V )  <->  ( D  e.  ( fBas `  Y
)  /\  A. v  e.  V  E. b  e.  D  ( b  X.  b )  C_  v
) ) )
7313, 70, 72mpbir2and 922 1  |-  ( ph  ->  D  e.  (CauFilu `  V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   `'ccnv 4988   ran crn 4990   "cima 4992   Fun wfun 5572    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   fBascfbas 18385  UnifOncust 20680   Cnucucn 20756  CauFiluccfilu 20767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-fbas 18395  df-ust 20681  df-ucn 20757  df-cfilu 20768
This theorem is referenced by:  ucnextcn  20785
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