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Theorem dfdm5 25392
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 1756 . . . 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 4419 . . . . . . . 8  |-  <. z ,  y >.  e.  _V
3 breq1 4207 . . . . . . . . . 10  |-  ( p  =  <. z ,  y
>.  ->  ( p 1st x  <->  <. z ,  y
>. 1st x ) )
4 eleq1 2495 . . . . . . . . . 10  |-  ( p  =  <. z ,  y
>.  ->  ( p  e.  A  <->  <. z ,  y
>.  e.  A ) )
53, 4anbi12d 692 . . . . . . . . 9  |-  ( p  =  <. z ,  y
>.  ->  ( ( p 1st x  /\  p  e.  A )  <->  ( <. z ,  y >. 1st x  /\  <. z ,  y
>.  e.  A ) ) )
6 vex 2951 . . . . . . . . . . . 12  |-  z  e. 
_V
7 vex 2951 . . . . . . . . . . . 12  |-  y  e. 
_V
8 vex 2951 . . . . . . . . . . . 12  |-  x  e. 
_V
96, 7, 8br1steq 25390 . . . . . . . . . . 11  |-  ( <.
z ,  y >. 1st x  <->  x  =  z
)
10 equcom 1692 . . . . . . . . . . 11  |-  ( x  =  z  <->  z  =  x )
119, 10bitri 241 . . . . . . . . . 10  |-  ( <.
z ,  y >. 1st x  <->  z  =  x )
1211anbi1i 677 . . . . . . . . 9  |-  ( (
<. z ,  y >. 1st x  /\  <. z ,  y >.  e.  A
)  <->  ( z  =  x  /\  <. z ,  y >.  e.  A
) )
135, 12syl6bb 253 . . . . . . . 8  |-  ( p  =  <. z ,  y
>.  ->  ( ( p 1st x  /\  p  e.  A )  <->  ( z  =  x  /\  <. z ,  y >.  e.  A
) ) )
142, 13ceqsexv 2983 . . . . . . 7  |-  ( E. p ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) )  <->  ( z  =  x  /\  <. z ,  y >.  e.  A
) )
1514exbii 1592 . . . . . 6  |-  ( E. z E. p ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
)  <->  E. z ( z  =  x  /\  <. z ,  y >.  e.  A
) )
16 excom 1756 . . . . . 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 3976 . . . . . . . 8  |-  ( z  =  x  ->  <. z ,  y >.  =  <. x ,  y >. )
1817eleq1d 2501 . . . . . . 7  |-  ( z  =  x  ->  ( <. z ,  y >.  e.  A  <->  <. x ,  y
>.  e.  A ) )
198, 18ceqsexv 2983 . . . . . 6  |-  ( E. z ( z  =  x  /\  <. z ,  y >.  e.  A
)  <->  <. x ,  y
>.  e.  A )
2015, 16, 193bitr3ri 268 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  E. p E. z
( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
) )
2120exbii 1592 . . . 4  |-  ( E. y <. x ,  y
>.  e.  A  <->  E. y E. p E. z ( p  =  <. z ,  y >.  /\  (
p 1st x  /\  p  e.  A )
) )
22 ancom 438 . . . . . 6  |-  ( ( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x )  <->  ( p
( 1st  |`  ( _V 
X.  _V ) ) x  /\  p  e.  A
) )
23 anass 631 . . . . . . 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 5144 . . . . . . . . 9  |-  ( p ( 1st  |`  ( _V  X.  _V ) ) x  <->  ( p 1st x  /\  p  e.  ( _V  X.  _V ) ) )
25 ancom 438 . . . . . . . . . 10  |-  ( ( p 1st x  /\  p  e.  ( _V  X.  _V ) )  <->  ( p  e.  ( _V  X.  _V )  /\  p 1st x
) )
26 elvv 4928 . . . . . . . . . . . 12  |-  ( p  e.  ( _V  X.  _V )  <->  E. z E. y  p  =  <. z ,  y >. )
27 excom 1756 . . . . . . . . . . . 12  |-  ( E. z E. y  p  =  <. z ,  y
>. 
<->  E. y E. z  p  =  <. z ,  y >. )
2826, 27bitri 241 . . . . . . . . . . 11  |-  ( p  e.  ( _V  X.  _V )  <->  E. y E. z  p  =  <. z ,  y >. )
2928anbi1i 677 . . . . . . . . . 10  |-  ( ( p  e.  ( _V 
X.  _V )  /\  p 1st x )  <->  ( E. y E. z  p  = 
<. z ,  y >.  /\  p 1st x ) )
3025, 29bitri 241 . . . . . . . . 9  |-  ( ( p 1st x  /\  p  e.  ( _V  X.  _V ) )  <->  ( E. y E. z  p  = 
<. z ,  y >.  /\  p 1st x ) )
3124, 30bitri 241 . . . . . . . 8  |-  ( p ( 1st  |`  ( _V  X.  _V ) ) x  <->  ( E. y E. z  p  =  <. z ,  y >.  /\  p 1st x ) )
3231anbi1i 677 . . . . . . 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 1925 . . . . . . 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 269 . . . . . 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 241 . . . . 5  |-  ( ( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x )  <->  E. y E. z ( p  = 
<. z ,  y >.  /\  ( p 1st x  /\  p  e.  A
) ) )
3635exbii 1592 . . . 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 269 . . 3  |-  ( E. y <. x ,  y
>.  e.  A  <->  E. p
( p  e.  A  /\  p ( 1st  |`  ( _V  X.  _V ) ) x ) )
388eldm2 5060 . . 3  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
398elima2 5201 . . 3  |-  ( x  e.  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)  <->  E. p ( p  e.  A  /\  p
( 1st  |`  ( _V 
X.  _V ) ) x ) )
4037, 38, 393bitr4i 269 . 2  |-  ( x  e.  dom  A  <->  x  e.  ( ( 1st  |`  ( _V  X.  _V ) )
" A ) )
4140eqriv 2432 1  |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204    X. cxp 4868   dom cdm 4870    |` cres 4872   "cima 4873   1stc1st 6339
This theorem is referenced by:  brdomain  25770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-1st 6341
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