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Theorem dfrn5 25402
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 1757 . . . 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 4428 . . . . . . . 8  |-  <. y ,  z >.  e.  _V
3 breq1 4216 . . . . . . . . . 10  |-  ( p  =  <. y ,  z
>.  ->  ( p 2nd x  <->  <. y ,  z
>. 2nd x ) )
4 eleq1 2497 . . . . . . . . . 10  |-  ( p  =  <. y ,  z
>.  ->  ( p  e.  A  <->  <. y ,  z
>.  e.  A ) )
53, 4anbi12d 693 . . . . . . . . 9  |-  ( p  =  <. y ,  z
>.  ->  ( ( p 2nd x  /\  p  e.  A )  <->  ( <. y ,  z >. 2nd x  /\  <. y ,  z
>.  e.  A ) ) )
6 vex 2960 . . . . . . . . . . . 12  |-  y  e. 
_V
7 vex 2960 . . . . . . . . . . . 12  |-  z  e. 
_V
8 vex 2960 . . . . . . . . . . . 12  |-  x  e. 
_V
96, 7, 8br2ndeq 25400 . . . . . . . . . . 11  |-  ( <.
y ,  z >. 2nd x  <->  x  =  z
)
10 equcom 1693 . . . . . . . . . . 11  |-  ( x  =  z  <->  z  =  x )
119, 10bitri 242 . . . . . . . . . 10  |-  ( <.
y ,  z >. 2nd x  <->  z  =  x )
1211anbi1i 678 . . . . . . . . 9  |-  ( (
<. y ,  z >. 2nd x  /\  <. y ,  z >.  e.  A
)  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) )
135, 12syl6bb 254 . . . . . . . 8  |-  ( p  =  <. y ,  z
>.  ->  ( ( p 2nd x  /\  p  e.  A )  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) ) )
142, 13ceqsexv 2992 . . . . . . 7  |-  ( E. p ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) )  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) )
1514exbii 1593 . . . . . 6  |-  ( E. z E. p ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
)  <->  E. z ( z  =  x  /\  <. y ,  z >.  e.  A
) )
16 excom 1757 . . . . . 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 3986 . . . . . . . 8  |-  ( z  =  x  ->  <. y ,  z >.  =  <. y ,  x >. )
1817eleq1d 2503 . . . . . . 7  |-  ( z  =  x  ->  ( <. y ,  z >.  e.  A  <->  <. y ,  x >.  e.  A ) )
198, 18ceqsexv 2992 . . . . . 6  |-  ( E. z ( z  =  x  /\  <. y ,  z >.  e.  A
)  <->  <. y ,  x >.  e.  A )
2015, 16, 193bitr3ri 269 . . . . 5  |-  ( <.
y ,  x >.  e.  A  <->  E. p E. z
( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
2120exbii 1593 . . . 4  |-  ( E. y <. y ,  x >.  e.  A  <->  E. y E. p E. z ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
22 ancom 439 . . . . . 6  |-  ( ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  ( p
( 2nd  |`  ( _V 
X.  _V ) ) x  /\  p  e.  A
) )
23 anass 632 . . . . . . 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 5153 . . . . . . . . 9  |-  ( p ( 2nd  |`  ( _V  X.  _V ) ) x  <->  ( p 2nd x  /\  p  e.  ( _V  X.  _V ) ) )
25 ancom 439 . . . . . . . . . 10  |-  ( ( p 2nd x  /\  p  e.  ( _V  X.  _V ) )  <->  ( p  e.  ( _V  X.  _V )  /\  p 2nd x
) )
26 elvv 4937 . . . . . . . . . . 11  |-  ( p  e.  ( _V  X.  _V )  <->  E. y E. z  p  =  <. y ,  z >. )
2726anbi1i 678 . . . . . . . . . 10  |-  ( ( p  e.  ( _V 
X.  _V )  /\  p 2nd x )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  p 2nd x ) )
2825, 27bitri 242 . . . . . . . . 9  |-  ( ( p 2nd x  /\  p  e.  ( _V  X.  _V ) )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  p 2nd x ) )
2924, 28bitri 242 . . . . . . . 8  |-  ( p ( 2nd  |`  ( _V  X.  _V ) ) x  <->  ( E. y E. z  p  =  <. y ,  z >.  /\  p 2nd x ) )
3029anbi1i 678 . . . . . . 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 1926 . . . . . . 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 270 . . . . . 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 242 . . . . 5  |-  ( ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  E. y E. z ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
3433exbii 1593 . . . 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 270 . . 3  |-  ( E. y <. y ,  x >.  e.  A  <->  E. p
( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x ) )
368elrn2 5110 . . 3  |-  ( x  e.  ran  A  <->  E. y <. y ,  x >.  e.  A )
378elima2 5210 . . 3  |-  ( x  e.  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)  <->  E. p ( p  e.  A  /\  p
( 2nd  |`  ( _V 
X.  _V ) ) x ) )
3835, 36, 373bitr4i 270 . 2  |-  ( x  e.  ran  A  <->  x  e.  ( ( 2nd  |`  ( _V  X.  _V ) )
" A ) )
3938eqriv 2434 1  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2957   <.cop 3818   class class class wbr 4213    X. cxp 4877   ran crn 4880    |` cres 4881   "cima 4882   2ndc2nd 6349
This theorem is referenced by:  brrange  25780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463  df-2nd 6351
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