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Theorem dfrn5 30426
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 1903 . . . 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 4685 . . . . . . . 8  |-  <. y ,  z >.  e.  _V
3 breq1 4426 . . . . . . . . . 10  |-  ( p  =  <. y ,  z
>.  ->  ( p 2nd x  <->  <. y ,  z
>. 2nd x ) )
4 eleq1 2495 . . . . . . . . . 10  |-  ( p  =  <. y ,  z
>.  ->  ( p  e.  A  <->  <. y ,  z
>.  e.  A ) )
53, 4anbi12d 715 . . . . . . . . 9  |-  ( p  =  <. y ,  z
>.  ->  ( ( p 2nd x  /\  p  e.  A )  <->  ( <. y ,  z >. 2nd x  /\  <. y ,  z
>.  e.  A ) ) )
6 vex 3083 . . . . . . . . . . . 12  |-  y  e. 
_V
7 vex 3083 . . . . . . . . . . . 12  |-  z  e. 
_V
8 vex 3083 . . . . . . . . . . . 12  |-  x  e. 
_V
96, 7, 8br2ndeq 30422 . . . . . . . . . . 11  |-  ( <.
y ,  z >. 2nd x  <->  x  =  z
)
10 equcom 1848 . . . . . . . . . . 11  |-  ( x  =  z  <->  z  =  x )
119, 10bitri 252 . . . . . . . . . 10  |-  ( <.
y ,  z >. 2nd x  <->  z  =  x )
1211anbi1i 699 . . . . . . . . 9  |-  ( (
<. y ,  z >. 2nd x  /\  <. y ,  z >.  e.  A
)  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) )
135, 12syl6bb 264 . . . . . . . 8  |-  ( p  =  <. y ,  z
>.  ->  ( ( p 2nd x  /\  p  e.  A )  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) ) )
142, 13ceqsexv 3118 . . . . . . 7  |-  ( E. p ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) )  <->  ( z  =  x  /\  <. y ,  z >.  e.  A
) )
1514exbii 1712 . . . . . 6  |-  ( E. z E. p ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
)  <->  E. z ( z  =  x  /\  <. y ,  z >.  e.  A
) )
16 excom 1903 . . . . . 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 4188 . . . . . . . 8  |-  ( z  =  x  ->  <. y ,  z >.  =  <. y ,  x >. )
1817eleq1d 2491 . . . . . . 7  |-  ( z  =  x  ->  ( <. y ,  z >.  e.  A  <->  <. y ,  x >.  e.  A ) )
198, 18ceqsexv 3118 . . . . . 6  |-  ( E. z ( z  =  x  /\  <. y ,  z >.  e.  A
)  <->  <. y ,  x >.  e.  A )
2015, 16, 193bitr3ri 279 . . . . 5  |-  ( <.
y ,  x >.  e.  A  <->  E. p E. z
( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
2120exbii 1712 . . . 4  |-  ( E. y <. y ,  x >.  e.  A  <->  E. y E. p E. z ( p  =  <. y ,  z >.  /\  (
p 2nd x  /\  p  e.  A )
) )
22 ancom 451 . . . . . 6  |-  ( ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  ( p
( 2nd  |`  ( _V 
X.  _V ) ) x  /\  p  e.  A
) )
23 anass 653 . . . . . . 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 5130 . . . . . . . . 9  |-  ( p ( 2nd  |`  ( _V  X.  _V ) ) x  <->  ( p 2nd x  /\  p  e.  ( _V  X.  _V ) ) )
25 ancom 451 . . . . . . . . . 10  |-  ( ( p 2nd x  /\  p  e.  ( _V  X.  _V ) )  <->  ( p  e.  ( _V  X.  _V )  /\  p 2nd x
) )
26 elvv 4912 . . . . . . . . . . 11  |-  ( p  e.  ( _V  X.  _V )  <->  E. y E. z  p  =  <. y ,  z >. )
2726anbi1i 699 . . . . . . . . . 10  |-  ( ( p  e.  ( _V 
X.  _V )  /\  p 2nd x )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  p 2nd x ) )
2825, 27bitri 252 . . . . . . . . 9  |-  ( ( p 2nd x  /\  p  e.  ( _V  X.  _V ) )  <->  ( E. y E. z  p  = 
<. y ,  z >.  /\  p 2nd x ) )
2924, 28bitri 252 . . . . . . . 8  |-  ( p ( 2nd  |`  ( _V  X.  _V ) ) x  <->  ( E. y E. z  p  =  <. y ,  z >.  /\  p 2nd x ) )
3029anbi1i 699 . . . . . . 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 1824 . . . . . . 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 280 . . . . . 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 252 . . . . 5  |-  ( ( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x )  <->  E. y E. z ( p  = 
<. y ,  z >.  /\  ( p 2nd x  /\  p  e.  A
) ) )
3433exbii 1712 . . . 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 280 . . 3  |-  ( E. y <. y ,  x >.  e.  A  <->  E. p
( p  e.  A  /\  p ( 2nd  |`  ( _V  X.  _V ) ) x ) )
368elrn2 5093 . . 3  |-  ( x  e.  ran  A  <->  E. y <. y ,  x >.  e.  A )
378elima2 5193 . . 3  |-  ( x  e.  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)  <->  E. p ( p  e.  A  /\  p
( 2nd  |`  ( _V 
X.  _V ) ) x ) )
3835, 36, 373bitr4i 280 . 2  |-  ( x  e.  ran  A  <->  x  e.  ( ( 2nd  |`  ( _V  X.  _V ) )
" A ) )
3938eqriv 2418 1  |-  ran  A  =  ( ( 2nd  |`  ( _V  X.  _V ) ) " A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080   <.cop 4004   class class class wbr 4423    X. cxp 4851   ran crn 4854    |` cres 4855   "cima 4856   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-2nd 6808
This theorem is referenced by:  brrange  30706
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