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Theorem brrestrict 24559
Description: The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brrestrict.1  |-  A  e. 
_V
brrestrict.2  |-  B  e. 
_V
brrestrict.3  |-  C  e. 
_V
Assertion
Ref Expression
brrestrict  |-  ( <. A ,  B >.Restrict C  <->  C  =  ( A  |`  B ) )

Proof of Theorem brrestrict
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4253 . . . . 5  |-  <. A ,  B >.  e.  _V
2 brrestrict.3 . . . . 5  |-  C  e. 
_V
31, 2brco 4868 . . . 4  |-  ( <. A ,  B >. (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) ) C  <->  E. x ( <. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  /\  xCap C ) )
41brtxp2 24492 . . . . . . 7  |-  ( <. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  <->  E. a E. b
( x  =  <. a ,  b >.  /\  <. A ,  B >. 1st a  /\  <. A ,  B >. (Cart  o.  ( 2nd  (x)  (Range  o.  1st )
) ) b ) )
5 3anrot 939 . . . . . . . . 9  |-  ( ( x  =  <. a ,  b >.  /\  <. A ,  B >. 1st a  /\  <. A ,  B >. (Cart  o.  ( 2nd  (x)  (Range  o.  1st )
) ) b )  <-> 
( <. A ,  B >. 1st a  /\  <. A ,  B >. (Cart  o.  ( 2nd  (x)  (Range  o. 
1st ) ) ) b  /\  x  = 
<. a ,  b >.
) )
6 brrestrict.1 . . . . . . . . . . 11  |-  A  e. 
_V
7 brrestrict.2 . . . . . . . . . . 11  |-  B  e. 
_V
8 vex 2804 . . . . . . . . . . 11  |-  a  e. 
_V
96, 7, 8br1steq 24201 . . . . . . . . . 10  |-  ( <. A ,  B >. 1st a  <->  a  =  A )
10 vex 2804 . . . . . . . . . . . 12  |-  b  e. 
_V
111, 10brco 4868 . . . . . . . . . . 11  |-  ( <. A ,  B >. (Cart 
o.  ( 2nd  (x)  (Range  o.  1st ) ) ) b  <->  E. x
( <. A ,  B >. ( 2nd  (x)  (Range  o. 
1st ) ) x  /\  xCart b ) )
121brtxp2 24492 . . . . . . . . . . . . . . 15  |-  ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  <->  E. a E. b ( x  = 
<. a ,  b >.  /\  <. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o. 
1st ) b ) )
13 3anrot 939 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  <. a ,  b >.  /\  <. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st )
b )  <->  ( <. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st )
b  /\  x  =  <. a ,  b >.
) )
146, 7, 8br2ndeq 24202 . . . . . . . . . . . . . . . . . 18  |-  ( <. A ,  B >. 2nd a  <->  a  =  B )
151, 10brco 4868 . . . . . . . . . . . . . . . . . . 19  |-  ( <. A ,  B >. (Range 
o.  1st ) b  <->  E. x
( <. A ,  B >. 1st x  /\  xRange b ) )
16 vex 2804 . . . . . . . . . . . . . . . . . . . . . . 23  |-  x  e. 
_V
176, 7, 16br1steq 24201 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. A ,  B >. 1st x  <->  x  =  A
)
1817anbi1i 676 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. A ,  B >. 1st x  /\  xRange b
)  <->  ( x  =  A  /\  xRange b
) )
1918exbii 1572 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. x ( <. A ,  B >. 1st x  /\  xRange b )  <->  E. x
( x  =  A  /\  xRange b ) )
20 breq1 4042 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  A  ->  (
xRange b  <->  ARange b
) )
216, 20ceqsexv 2836 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. x ( x  =  A  /\  xRange b
)  <->  ARange b )
2219, 21bitri 240 . . . . . . . . . . . . . . . . . . 19  |-  ( E. x ( <. A ,  B >. 1st x  /\  xRange b )  <->  ARange b
)
236, 10brrange 24544 . . . . . . . . . . . . . . . . . . 19  |-  ( ARange b  <->  b  =  ran  A )
2415, 22, 233bitri 262 . . . . . . . . . . . . . . . . . 18  |-  ( <. A ,  B >. (Range 
o.  1st ) b  <->  b  =  ran  A )
25 biid 227 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. a ,  b
>. 
<->  x  =  <. a ,  b >. )
2614, 24, 253anbi123i 1140 . . . . . . . . . . . . . . . . 17  |-  ( (
<. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st ) b  /\  x  =  <. a ,  b
>. )  <->  ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b
>. ) )
2713, 26bitri 240 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  <. a ,  b >.  /\  <. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st )
b )  <->  ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b
>. ) )
28272exbii 1573 . . . . . . . . . . . . . . 15  |-  ( E. a E. b ( x  =  <. a ,  b >.  /\  <. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st )
b )  <->  E. a E. b ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b
>. ) )
296rnex 4958 . . . . . . . . . . . . . . . 16  |-  ran  A  e.  _V
30 opeq1 3812 . . . . . . . . . . . . . . . . 17  |-  ( a  =  B  ->  <. a ,  b >.  =  <. B ,  b >. )
3130eqeq2d 2307 . . . . . . . . . . . . . . . 16  |-  ( a  =  B  ->  (
x  =  <. a ,  b >.  <->  x  =  <. B ,  b >.
) )
32 opeq2 3813 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ran  A  ->  <. B ,  b >.  =  <. B ,  ran  A
>. )
3332eqeq2d 2307 . . . . . . . . . . . . . . . 16  |-  ( b  =  ran  A  -> 
( x  =  <. B ,  b >.  <->  x  =  <. B ,  ran  A >. ) )
347, 29, 31, 33ceqsex2v 2838 . . . . . . . . . . . . . . 15  |-  ( E. a E. b ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b >. )  <->  x  =  <. B ,  ran  A >. )
3512, 28, 343bitri 262 . . . . . . . . . . . . . 14  |-  ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  <->  x  =  <. B ,  ran  A >. )
3635anbi1i 676 . . . . . . . . . . . . 13  |-  ( (
<. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  /\  xCart b )  <->  ( x  =  <. B ,  ran  A
>.  /\  xCart b ) )
3736exbii 1572 . . . . . . . . . . . 12  |-  ( E. x ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  /\  xCart b
)  <->  E. x ( x  =  <. B ,  ran  A
>.  /\  xCart b ) )
38 opex 4253 . . . . . . . . . . . . 13  |-  <. B ,  ran  A >.  e.  _V
39 breq1 4042 . . . . . . . . . . . . 13  |-  ( x  =  <. B ,  ran  A
>.  ->  ( xCart b  <->  <. B ,  ran  A >.Cart b ) )
4038, 39ceqsexv 2836 . . . . . . . . . . . 12  |-  ( E. x ( x  = 
<. B ,  ran  A >.  /\  xCart b )  <->  <. B ,  ran  A >.Cart b )
4137, 40bitri 240 . . . . . . . . . . 11  |-  ( E. x ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  /\  xCart b
)  <->  <. B ,  ran  A
>.Cart b )
427, 29, 10brcart 24542 . . . . . . . . . . 11  |-  ( <. B ,  ran  A >.Cart b  <-> 
b  =  ( B  X.  ran  A ) )
4311, 41, 423bitri 262 . . . . . . . . . 10  |-  ( <. A ,  B >. (Cart 
o.  ( 2nd  (x)  (Range  o.  1st ) ) ) b  <->  b  =  ( B  X.  ran  A
) )
449, 43, 253anbi123i 1140 . . . . . . . . 9  |-  ( (
<. A ,  B >. 1st a  /\  <. A ,  B >. (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) b  /\  x  =  <. a ,  b >. )  <->  ( a  =  A  /\  b  =  ( B  X.  ran  A )  /\  x  =  <. a ,  b >. ) )
455, 44bitri 240 . . . . . . . 8  |-  ( ( x  =  <. a ,  b >.  /\  <. A ,  B >. 1st a  /\  <. A ,  B >. (Cart  o.  ( 2nd  (x)  (Range  o.  1st )
) ) b )  <-> 
( a  =  A  /\  b  =  ( B  X.  ran  A
)  /\  x  =  <. a ,  b >.
) )
46452exbii 1573 . . . . . . 7  |-  ( E. a E. b ( x  =  <. a ,  b >.  /\  <. A ,  B >. 1st a  /\  <. A ,  B >. (Cart  o.  ( 2nd  (x)  (Range  o.  1st )
) ) b )  <->  E. a E. b ( a  =  A  /\  b  =  ( B  X.  ran  A )  /\  x  =  <. a ,  b >. ) )
477, 29xpex 4817 . . . . . . . 8  |-  ( B  X.  ran  A )  e.  _V
48 opeq1 3812 . . . . . . . . 9  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
4948eqeq2d 2307 . . . . . . . 8  |-  ( a  =  A  ->  (
x  =  <. a ,  b >.  <->  x  =  <. A ,  b >.
) )
50 opeq2 3813 . . . . . . . . 9  |-  ( b  =  ( B  X.  ran  A )  ->  <. A , 
b >.  =  <. A , 
( B  X.  ran  A ) >. )
5150eqeq2d 2307 . . . . . . . 8  |-  ( b  =  ( B  X.  ran  A )  ->  (
x  =  <. A , 
b >. 
<->  x  =  <. A , 
( B  X.  ran  A ) >. ) )
526, 47, 49, 51ceqsex2v 2838 . . . . . . 7  |-  ( E. a E. b ( a  =  A  /\  b  =  ( B  X.  ran  A )  /\  x  =  <. a ,  b >. )  <->  x  =  <. A ,  ( B  X.  ran  A )
>. )
534, 46, 523bitri 262 . . . . . 6  |-  ( <. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  <->  x  =  <. A ,  ( B  X.  ran  A ) >. )
5453anbi1i 676 . . . . 5  |-  ( (
<. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  /\  xCap C
)  <->  ( x  = 
<. A ,  ( B  X.  ran  A )
>.  /\  xCap C ) )
5554exbii 1572 . . . 4  |-  ( E. x ( <. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  /\  xCap C )  <->  E. x
( x  =  <. A ,  ( B  X.  ran  A ) >.  /\  xCap C ) )
563, 55bitri 240 . . 3  |-  ( <. A ,  B >. (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) ) C  <->  E. x ( x  = 
<. A ,  ( B  X.  ran  A )
>.  /\  xCap C ) )
57 opex 4253 . . . 4  |-  <. A , 
( B  X.  ran  A ) >.  e.  _V
58 breq1 4042 . . . 4  |-  ( x  =  <. A ,  ( B  X.  ran  A
) >.  ->  ( xCap C 
<-> 
<. A ,  ( B  X.  ran  A )
>.Cap C ) )
5957, 58ceqsexv 2836 . . 3  |-  ( E. x ( x  = 
<. A ,  ( B  X.  ran  A )
>.  /\  xCap C )  <->  <. A ,  ( B  X.  ran  A )
>.Cap C )
606, 47, 2brcap 24550 . . 3  |-  ( <. A ,  ( B  X.  ran  A ) >.Cap C 
<->  C  =  ( A  i^i  ( B  X.  ran  A ) ) )
6156, 59, 603bitri 262 . 2  |-  ( <. A ,  B >. (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) ) C  <-> 
C  =  ( A  i^i  ( B  X.  ran  A ) ) )
62 df-restrict 24483 . . 3  |- Restrict  =  (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) )
6362breqi 4045 . 2  |-  ( <. A ,  B >.Restrict C  <->  <. A ,  B >. (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) ) C )
64 dfres3 24187 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A ) )
6564eqeq2i 2306 . 2  |-  ( C  =  ( A  |`  B )  <->  C  =  ( A  i^i  ( B  X.  ran  A ) ) )
6661, 63, 653bitr4i 268 1  |-  ( <. A ,  B >.Restrict C  <->  C  =  ( A  |`  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   <.cop 3656   class class class wbr 4039    X. cxp 4703   ran crn 4706    |` cres 4707    o. ccom 4709   1stc1st 6136   2ndc2nd 6137    (x) ctxp 24444  Cartccart 24455  Rangecrange 24458  Capccap 24461  Restrictcrestrict 24465
This theorem is referenced by:  tfrqfree  24561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-pprod 24467  df-image 24476  df-cart 24477  df-range 24480  df-cap 24482  df-restrict 24483
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