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Theorem dfdm5 24203
Description: Definition of domain in terms of  1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfdm5  |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)

Proof of Theorem dfdm5
Dummy variables  p  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 1798 . . . 4  |-  ( E. y E. p E. z ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) )  <->  E. p E. y E. z ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
) )
2 opex 4253 . . . . . . . 8  |-  <. z ,  y >.  e.  _V
3 breq1 4042 . . . . . . . . . 10  |-  ( p  =  <. z ,  y
>.  ->  ( p 1st x  <->  <. z ,  y
>. 1st x ) )
4 eleq1 2356 . . . . . . . . . 10  |-  ( p  =  <. z ,  y
>.  ->  ( p  e.  A  <->  <. z ,  y
>.  e.  A ) )
53, 4anbi12d 691 . . . . . . . . 9  |-  ( p  =  <. z ,  y
>.  ->  ( ( p 1st x  /\  p  e.  A )  <->  ( <. z ,  y >. 1st x  /\  <. z ,  y
>.  e.  A ) ) )
6 vex 2804 . . . . . . . . . . . 12  |-  z  e. 
_V
7 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
8 vex 2804 . . . . . . . . . . . 12  |-  x  e. 
_V
96, 7, 8br1steq 24201 . . . . . . . . . . 11  |-  ( <.
z ,  y >. 1st x  <->  x  =  z
)
10 equcom 1665 . . . . . . . . . . 11  |-  ( x  =  z  <->  z  =  x )
119, 10bitri 240 . . . . . . . . . 10  |-  ( <.
z ,  y >. 1st x  <->  z  =  x )
1211anbi1i 676 . . . . . . . . 9  |-  ( (
<. z ,  y >. 1st x  /\  <. z ,  y >.  e.  A
)  <->  ( z  =  x  /\  <. z ,  y >.  e.  A
) )
135, 12syl6bb 252 . . . . . . . 8  |-  ( p  =  <. z ,  y
>.  ->  ( ( p 1st x  /\  p  e.  A )  <->  ( z  =  x  /\  <. z ,  y >.  e.  A
) ) )
142, 13ceqsexv 2836 . . . . . . 7  |-  ( E. p ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) )  <->  ( z  =  x  /\  <. z ,  y >.  e.  A
) )
1514exbii 1572 . . . . . 6  |-  ( E. z E. p ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
)  <->  E. z ( z  =  x  /\  <. z ,  y >.  e.  A
) )
16 excom 1798 . . . . . 6  |-  ( E. z E. p ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
)  <->  E. p E. z
( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
) )
17 opeq1 3812 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  y >.  =  <. x ,  y >. )
1817eleq1d 2362 . . . . . . 7  |-  ( z  =  x  ->  ( <. z ,  y >.  e.  A  <->  <. x ,  y
>.  e.  A ) )
198, 18ceqsexv 2836 . . . . . 6  |-  ( E. z ( z  =  x  /\  <. z ,  y >.  e.  A
)  <->  <. x ,  y
>.  e.  A )
2015, 16, 193bitr3ri 267 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  E. p E. z
( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
) )
2120exbii 1572 . . . 4  |-  ( E. y <. x ,  y
>.  e.  A  <->  E. y E. p E. z ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
) )
22 ancom 437 . . . . . 6  |-  ( ( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x )  <->  ( p
( 1st  |`  ( _V 
X.  _V ) ) x  /\  p  e.  A
) )
23 anass 630 . . . . . . 7  |-  ( ( ( E. y E. z  p  =  <. z ,  y >.  /\  p 1st x )  /\  p  e.  A )  <->  ( E. y E. z  p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) ) )
248brres 4977 . . . . . . . . 9  |-  ( p ( 1st  |`  ( _V  X.  _V ) ) x  <->  ( p 1st x  /\  p  e.  ( _V  X.  _V ) ) )
25 ancom 437 . . . . . . . . . 10  |-  ( ( p 1st x  /\  p  e.  ( _V  X.  _V ) )  <->  ( p  e.  ( _V  X.  _V )  /\  p 1st x
) )
26 elvv 4764 . . . . . . . . . . . 12  |-  ( p  e.  ( _V  X.  _V )  <->  E. z E. y  p  =  <. z ,  y >. )
27 excom 1798 . . . . . . . . . . . 12  |-  ( E. z E. y  p  =  <. z ,  y
>. 
<->  E. y E. z  p  =  <. z ,  y >. )
2826, 27bitri 240 . . . . . . . . . . 11  |-  ( p  e.  ( _V  X.  _V )  <->  E. y E. z  p  =  <. z ,  y >. )
2928anbi1i 676 . . . . . . . . . 10  |-  ( ( p  e.  ( _V 
X.  _V )  /\  p 1st x )  <->  ( E. y E. z  p  = 
<. z ,  y >.  /\  p 1st x ) )
3025, 29bitri 240 . . . . . . . . 9  |-  ( ( p 1st x  /\  p  e.  ( _V  X.  _V ) )  <->  ( E. y E. z  p  = 
<. z ,  y >.  /\  p 1st x ) )
3124, 30bitri 240 . . . . . . . 8  |-  ( p ( 1st  |`  ( _V  X.  _V ) ) x  <->  ( E. y E. z  p  =  <. z ,  y >.  /\  p 1st x ) )
3231anbi1i 676 . . . . . . 7  |-  ( ( p ( 1st  |`  ( _V  X.  _V ) ) x  /\  p  e.  A )  <->  ( ( E. y E. z  p  =  <. z ,  y
>.  /\  p 1st x
)  /\  p  e.  A ) )
33 19.41vv 1855 . . . . . . 7  |-  ( E. y E. z ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
)  <->  ( E. y E. z  p  =  <. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) ) )
3423, 32, 333bitr4i 268 . . . . . 6  |-  ( ( p ( 1st  |`  ( _V  X.  _V ) ) x  /\  p  e.  A )  <->  E. y E. z ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) ) )
3522, 34bitri 240 . . . . 5  |-  ( ( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x )  <->  E. y E. z ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) ) )
3635exbii 1572 . . . 4  |-  ( E. p ( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x )  <->  E. p E. y E. z ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) ) )
371, 21, 363bitr4i 268 . . 3  |-  ( E. y <. x ,  y
>.  e.  A  <->  E. p
( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x ) )
388eldm2 4893 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
398elima2 5034 . . 3  |-  ( x  e.  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)  <->  E. p ( p  e.  A  /\  p
( 1st  |`  ( _V 
X.  _V ) ) x ) )
4037, 38, 393bitr4i 268 . 2  |-  ( x  e.  dom  A  <->  x  e.  ( ( 1st  |`  ( _V  X.  _V ) )
" A ) )
4140eqriv 2293 1  |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705    |` cres 4707   "cima 4708   1stc1st 6136
This theorem is referenced by:  brdomain  24543
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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  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
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