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Theorem brrestrict 27985
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 4561 . . . . 5  |-  <. A ,  B >.  e.  _V
2 brrestrict.3 . . . . 5  |-  C  e. 
_V
31, 2brco 5015 . . . 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 27917 . . . . . . 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 970 . . . . . . . . 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 2980 . . . . . . . . . . 11  |-  a  e. 
_V
96, 7, 8br1steq 27590 . . . . . . . . . 10  |-  ( <. A ,  B >. 1st a  <->  a  =  A )
10 vex 2980 . . . . . . . . . . . 12  |-  b  e. 
_V
111, 10brco 5015 . . . . . . . . . . 11  |-  ( <. A ,  B >. (Cart 
o.  ( 2nd  (x)  (Range  o.  1st ) ) ) b  <->  E. x
( <. A ,  B >. ( 2nd  (x)  (Range  o. 
1st ) ) x  /\  xCart b ) )
121brtxp2 27917 . . . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . . . . 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 27591 . . . . . . . . . . . . . . . . . 18  |-  ( <. A ,  B >. 2nd a  <->  a  =  B )
151, 10brco 5015 . . . . . . . . . . . . . . . . . . 19  |-  ( <. A ,  B >. (Range 
o.  1st ) b  <->  E. x
( <. A ,  B >. 1st x  /\  xRange b ) )
16 vex 2980 . . . . . . . . . . . . . . . . . . . . . . 23  |-  x  e. 
_V
176, 7, 16br1steq 27590 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. A ,  B >. 1st x  <->  x  =  A
)
1817anbi1i 695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. A ,  B >. 1st x  /\  xRange b
)  <->  ( x  =  A  /\  xRange b
) )
1918exbii 1634 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. x ( <. A ,  B >. 1st x  /\  xRange b )  <->  E. x
( x  =  A  /\  xRange b ) )
20 breq1 4300 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  A  ->  (
xRange b  <->  ARange b
) )
216, 20ceqsexv 3014 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. x ( x  =  A  /\  xRange b
)  <->  ARange b )
2219, 21bitri 249 . . . . . . . . . . . . . . . . . . 19  |-  ( E. x ( <. A ,  B >. 1st x  /\  xRange b )  <->  ARange b
)
236, 10brrange 27970 . . . . . . . . . . . . . . . . . . 19  |-  ( ARange b  <->  b  =  ran  A )
2415, 22, 233bitri 271 . . . . . . . . . . . . . . . . . 18  |-  ( <. A ,  B >. (Range 
o.  1st ) b  <->  b  =  ran  A )
25 biid 236 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. a ,  b
>. 
<->  x  =  <. a ,  b >. )
2614, 24, 253anbi123i 1176 . . . . . . . . . . . . . . . . 17  |-  ( (
<. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st ) b  /\  x  =  <. a ,  b
>. )  <->  ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b
>. ) )
2713, 26bitri 249 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  <. a ,  b >.  /\  <. A ,  B >. 2nd a  /\  <. A ,  B >. (Range  o.  1st )
b )  <->  ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b
>. ) )
28272exbii 1635 . . . . . . . . . . . . . . 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 6517 . . . . . . . . . . . . . . . 16  |-  ran  A  e.  _V
30 opeq1 4064 . . . . . . . . . . . . . . . . 17  |-  ( a  =  B  ->  <. a ,  b >.  =  <. B ,  b >. )
3130eqeq2d 2454 . . . . . . . . . . . . . . . 16  |-  ( a  =  B  ->  (
x  =  <. a ,  b >.  <->  x  =  <. B ,  b >.
) )
32 opeq2 4065 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ran  A  ->  <. B ,  b >.  =  <. B ,  ran  A
>. )
3332eqeq2d 2454 . . . . . . . . . . . . . . . 16  |-  ( b  =  ran  A  -> 
( x  =  <. B ,  b >.  <->  x  =  <. B ,  ran  A >. ) )
347, 29, 31, 33ceqsex2v 3016 . . . . . . . . . . . . . . 15  |-  ( E. a E. b ( a  =  B  /\  b  =  ran  A  /\  x  =  <. a ,  b >. )  <->  x  =  <. B ,  ran  A >. )
3512, 28, 343bitri 271 . . . . . . . . . . . . . 14  |-  ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  <->  x  =  <. B ,  ran  A >. )
3635anbi1i 695 . . . . . . . . . . . . 13  |-  ( (
<. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  /\  xCart b )  <->  ( x  =  <. B ,  ran  A
>.  /\  xCart b ) )
3736exbii 1634 . . . . . . . . . . . 12  |-  ( E. x ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  /\  xCart b
)  <->  E. x ( x  =  <. B ,  ran  A
>.  /\  xCart b ) )
38 opex 4561 . . . . . . . . . . . . 13  |-  <. B ,  ran  A >.  e.  _V
39 breq1 4300 . . . . . . . . . . . . 13  |-  ( x  =  <. B ,  ran  A
>.  ->  ( xCart b  <->  <. B ,  ran  A >.Cart b ) )
4038, 39ceqsexv 3014 . . . . . . . . . . . 12  |-  ( E. x ( x  = 
<. B ,  ran  A >.  /\  xCart b )  <->  <. B ,  ran  A >.Cart b )
4137, 40bitri 249 . . . . . . . . . . 11  |-  ( E. x ( <. A ,  B >. ( 2nd  (x)  (Range  o.  1st ) ) x  /\  xCart b
)  <->  <. B ,  ran  A
>.Cart b )
427, 29, 10brcart 27968 . . . . . . . . . . 11  |-  ( <. B ,  ran  A >.Cart b  <-> 
b  =  ( B  X.  ran  A ) )
4311, 41, 423bitri 271 . . . . . . . . . 10  |-  ( <. A ,  B >. (Cart 
o.  ( 2nd  (x)  (Range  o.  1st ) ) ) b  <->  b  =  ( B  X.  ran  A
) )
449, 43, 253anbi123i 1176 . . . . . . . . 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 249 . . . . . . . 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 1635 . . . . . . 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 6513 . . . . . . . 8  |-  ( B  X.  ran  A )  e.  _V
48 opeq1 4064 . . . . . . . . 9  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
4948eqeq2d 2454 . . . . . . . 8  |-  ( a  =  A  ->  (
x  =  <. a ,  b >.  <->  x  =  <. A ,  b >.
) )
50 opeq2 4065 . . . . . . . . 9  |-  ( b  =  ( B  X.  ran  A )  ->  <. A , 
b >.  =  <. A , 
( B  X.  ran  A ) >. )
5150eqeq2d 2454 . . . . . . . 8  |-  ( b  =  ( B  X.  ran  A )  ->  (
x  =  <. A , 
b >. 
<->  x  =  <. A , 
( B  X.  ran  A ) >. ) )
526, 47, 49, 51ceqsex2v 3016 . . . . . . 7  |-  ( E. a E. b ( a  =  A  /\  b  =  ( B  X.  ran  A )  /\  x  =  <. a ,  b >. )  <->  x  =  <. A ,  ( B  X.  ran  A )
>. )
534, 46, 523bitri 271 . . . . . 6  |-  ( <. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  <->  x  =  <. A ,  ( B  X.  ran  A ) >. )
5453anbi1i 695 . . . . 5  |-  ( (
<. A ,  B >. ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) x  /\  xCap C
)  <->  ( x  = 
<. A ,  ( B  X.  ran  A )
>.  /\  xCap C ) )
5554exbii 1634 . . . 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 249 . . 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 4561 . . . 4  |-  <. A , 
( B  X.  ran  A ) >.  e.  _V
58 breq1 4300 . . . 4  |-  ( x  =  <. A ,  ( B  X.  ran  A
) >.  ->  ( xCap C 
<-> 
<. A ,  ( B  X.  ran  A )
>.Cap C ) )
5957, 58ceqsexv 3014 . . 3  |-  ( E. x ( x  = 
<. A ,  ( B  X.  ran  A )
>.  /\  xCap C )  <->  <. A ,  ( B  X.  ran  A )
>.Cap C )
606, 47, 2brcap 27976 . . 3  |-  ( <. A ,  ( B  X.  ran  A ) >.Cap C 
<->  C  =  ( A  i^i  ( B  X.  ran  A ) ) )
6156, 59, 603bitri 271 . 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 27906 . . 3  |- Restrict  =  (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) )
6362breqi 4303 . 2  |-  ( <. A ,  B >.Restrict C  <->  <. A ,  B >. (Cap 
o.  ( 1st  (x)  (Cart  o.  ( 2nd  (x)  (Range  o.  1st ) ) ) ) ) C )
64 dfres3 27574 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A ) )
6564eqeq2i 2453 . 2  |-  ( C  =  ( A  |`  B )  <->  C  =  ( A  i^i  ( B  X.  ran  A ) ) )
6661, 63, 653bitr4i 277 1  |-  ( <. A ,  B >.Restrict C  <->  C  =  ( A  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2977    i^i cin 3332   <.cop 3888   class class class wbr 4297    X. cxp 4843   ran crn 4846    |` cres 4847    o. ccom 4849   1stc1st 6580   2ndc2nd 6581    (x) ctxp 27865  Cartccart 27876  Rangecrange 27879  Capccap 27882  Restrictcrestrict 27886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-eprel 4637  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fo 5429  df-fv 5431  df-1st 6582  df-2nd 6583  df-symdif 27854  df-txp 27889  df-pprod 27890  df-image 27899  df-cart 27900  df-range 27903  df-cap 27905  df-restrict 27906
This theorem is referenced by:  tfrqfree  27987
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