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Theorem dfrn5 24133
Description: Definition of range in terms of  2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfrn5  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)

Proof of Theorem dfrn5
Dummy variables  p  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 1786 . . . 4  |-  ( E. y E. p E. z ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) )  <->  E. p E. y E. z ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
2 opex 4237 . . . . . . . 8  |-  <. y ,  z >.  e.  _V
3 breq1 4026 . . . . . . . . . 10  |-  ( p  =  <. y ,  z
>.  ->  ( p 2nd x  <->  <. y ,  z
>. 2nd x ) )
4 eleq1 2343 . . . . . . . . . 10  |-  ( p  =  <. y ,  z
>.  ->  ( p  e.  A  <->  <. y ,  z
>.  e.  A ) )
53, 4anbi12d 691 . . . . . . . . 9  |-  ( p  =  <. y ,  z
>.  ->  ( ( p 2nd x  /\  p  e.  A )  <->  ( <. y ,  z >. 2nd x  /\  <. y ,  z
>.  e.  A ) ) )
6 vex 2791 . . . . . . . . . . . 12  |-  y  e. 
_V
7 vex 2791 . . . . . . . . . . . 12  |-  z  e. 
_V
8 vex 2791 . . . . . . . . . . . 12  |-  x  e. 
_V
96, 7, 8br2ndeq 24131 . . . . . . . . . . 11  |-  ( <.
y ,  z >. 2nd x  <->  x  =  z
)
10 equcom 1647 . . . . . . . . . . 11  |-  ( x  =  z  <->  z  =  x )
119, 10bitri 240 . . . . . . . . . 10  |-  ( <.
y ,  z >. 2nd x  <->  z  =  x )
1211anbi1i 676 . . . . . . . . 9  |-  ( (
<. y ,  z >. 2nd x  /\  <. y ,  z >.  e.  A
)  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) )
135, 12syl6bb 252 . . . . . . . 8  |-  ( p  =  <. y ,  z
>.  ->  ( ( p 2nd x  /\  p  e.  A )  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) ) )
142, 13ceqsexv 2823 . . . . . . 7  |-  ( E. p ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) )  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) )
1514exbii 1569 . . . . . 6  |-  ( E. z E. p ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
)  <->  E. z ( z  =  x  /\  <. y ,  z >.  e.  A
) )
16 excom 1786 . . . . . 6  |-  ( E. z E. p ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
)  <->  E. p E. z
( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
17 opeq2 3797 . . . . . . . 8  |-  ( z  =  x  ->  <. y ,  z >.  =  <. y ,  x >. )
1817eleq1d 2349 . . . . . . 7  |-  ( z  =  x  ->  ( <. y ,  z >.  e.  A  <->  <. y ,  x >.  e.  A ) )
198, 18ceqsexv 2823 . . . . . 6  |-  ( E. z ( z  =  x  /\  <. y ,  z >.  e.  A
)  <->  <. y ,  x >.  e.  A )
2015, 16, 193bitr3ri 267 . . . . 5  |-  ( <.
y ,  x >.  e.  A  <->  E. p E. z
( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
2120exbii 1569 . . . 4  |-  ( E. y <. y ,  x >.  e.  A  <->  E. y E. p E. z ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
22 ancom 437 . . . . . 6  |-  ( ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  ( p
( 2nd  |`  ( _V 
X.  _V ) ) x  /\  p  e.  A
) )
23 anass 630 . . . . . . 7  |-  ( ( ( E. y E. z  p  =  <. y ,  z >.  /\  p 2nd x )  /\  p  e.  A )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
248brres 4961 . . . . . . . . 9  |-  ( p ( 2nd  |`  ( _V  X.  _V ) ) x  <->  ( p 2nd x  /\  p  e.  ( _V  X.  _V ) ) )
25 ancom 437 . . . . . . . . . 10  |-  ( ( p 2nd x  /\  p  e.  ( _V  X.  _V ) )  <->  ( p  e.  ( _V  X.  _V )  /\  p 2nd x
) )
26 elvv 4748 . . . . . . . . . . 11  |-  ( p  e.  ( _V  X.  _V )  <->  E. y E. z  p  =  <. y ,  z >. )
2726anbi1i 676 . . . . . . . . . 10  |-  ( ( p  e.  ( _V 
X.  _V )  /\  p 2nd x )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  p 2nd x ) )
2825, 27bitri 240 . . . . . . . . 9  |-  ( ( p 2nd x  /\  p  e.  ( _V  X.  _V ) )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  p 2nd x ) )
2924, 28bitri 240 . . . . . . . 8  |-  ( p ( 2nd  |`  ( _V  X.  _V ) ) x  <->  ( E. y E. z  p  =  <. y ,  z >.  /\  p 2nd x ) )
3029anbi1i 676 . . . . . . 7  |-  ( ( p ( 2nd  |`  ( _V  X.  _V ) ) x  /\  p  e.  A )  <->  ( ( E. y E. z  p  =  <. y ,  z
>.  /\  p 2nd x
)  /\  p  e.  A ) )
31 19.41vv 1843 . . . . . . 7  |-  ( E. y E. z ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
)  <->  ( E. y E. z  p  =  <. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
3223, 30, 313bitr4i 268 . . . . . 6  |-  ( ( p ( 2nd  |`  ( _V  X.  _V ) ) x  /\  p  e.  A )  <->  E. y E. z ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
3322, 32bitri 240 . . . . 5  |-  ( ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  E. y E. z ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
3433exbii 1569 . . . 4  |-  ( E. p ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  E. p E. y E. z ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
351, 21, 343bitr4i 268 . . 3  |-  ( E. y <. y ,  x >.  e.  A  <->  E. p
( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x ) )
368elrn2 4918 . . 3  |-  ( x  e.  ran  A  <->  E. y <. y ,  x >.  e.  A )
378elima2 5018 . . 3  |-  ( x  e.  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)  <->  E. p ( p  e.  A  /\  p
( 2nd  |`  ( _V 
X.  _V ) ) x ) )
3835, 36, 373bitr4i 268 . 2  |-  ( x  e.  ran  A  <->  x  e.  ( ( 2nd  |`  ( _V  X.  _V ) )
" A ) )
3938eqriv 2280 1  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692   2ndc2nd 6121
This theorem is referenced by:  brrange  24473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-2nd 6123
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