Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elfuns Structured version   Unicode version

Theorem elfuns 27944
Description: Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypothesis
Ref Expression
elfuns.1  |-  F  e. 
_V
Assertion
Ref Expression
elfuns  |-  ( F  e.  Funs  <->  Fun  F )

Proof of Theorem elfuns
Dummy variables  a  x  y  z  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrel 4940 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  p  e.  F )  ->  E. x E. y  p  =  <. x ,  y >.
)
21ex 434 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( p  e.  F  ->  E. x E. y  p  =  <. x ,  y >.
) )
3 elrel 4940 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  q  e.  F )  ->  E. a E. z  q  =  <. a ,  z >.
)
43ex 434 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( q  e.  F  ->  E. a E. z  q  =  <. a ,  z >.
) )
52, 4anim12d 563 . . . . . . . . 9  |-  ( Rel 
F  ->  ( (
p  e.  F  /\  q  e.  F )  ->  ( E. x E. y  p  =  <. x ,  y >.  /\  E. a E. z  q  = 
<. a ,  z >.
) ) )
65adantrd 468 . . . . . . . 8  |-  ( Rel 
F  ->  ( (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )  ->  ( E. x E. y  p  = 
<. x ,  y >.  /\  E. a E. z 
q  =  <. a ,  z >. )
) )
76pm4.71rd 635 . . . . . . 7  |-  ( Rel 
F  ->  ( (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )  <->  ( ( E. x E. y  p  =  <. x ,  y
>.  /\  E. a E. z  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) ) )
8 19.41vvvv 1923 . . . . . . . 8  |-  ( E. x E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  ( E. x E. y E. a E. z ( p  = 
<. x ,  y >.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
9 ee4anv 1934 . . . . . . . . 9  |-  ( E. x E. y E. a E. z ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  <->  ( E. x E. y  p  =  <. x ,  y >.  /\  E. a E. z 
q  =  <. a ,  z >. )
)
109anbi1i 695 . . . . . . . 8  |-  ( ( E. x E. y E. a E. z ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  ( ( E. x E. y  p  =  <. x ,  y
>.  /\  E. a E. z  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
118, 10bitr2i 250 . . . . . . 7  |-  ( ( ( E. x E. y  p  =  <. x ,  y >.  /\  E. a E. z  q  = 
<. a ,  z >.
)  /\  ( (
p  e.  F  /\  q  e.  F )  /\  q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  E. x E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) ) )
127, 11syl6bb 261 . . . . . 6  |-  ( Rel 
F  ->  ( (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )  <->  E. x E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) ) ) )
13122exbidv 1682 . . . . 5  |-  ( Rel 
F  ->  ( E. p E. q ( ( p  e.  F  /\  q  e.  F )  /\  q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p )  <->  E. p E. q E. x E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) ) ) )
14 excom13 1789 . . . . . 6  |-  ( E. p E. q E. x E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  E. x E. q E. p E. y E. a E. z
( ( p  = 
<. x ,  y >.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
15 excom13 1789 . . . . . . . 8  |-  ( E. q E. p E. y E. a E. z
( ( p  = 
<. x ,  y >.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  E. y E. p E. q E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) ) )
16 exrot4 1791 . . . . . . . . . 10  |-  ( E. p E. q E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  E. a E. z E. p E. q ( ( p  =  <. x ,  y
>.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
17 excom 1787 . . . . . . . . . 10  |-  ( E. a E. z E. p E. q ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  E. z E. a E. p E. q ( ( p  =  <. x ,  y
>.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
18 df-3an 967 . . . . . . . . . . . . . . . 16  |-  ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>.  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  ( (
p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) ) )
19182exbii 1635 . . . . . . . . . . . . . . 15  |-  ( E. p E. q ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>.  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  E. p E. q ( ( p  =  <. x ,  y
>.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
20 opex 4554 . . . . . . . . . . . . . . . 16  |-  <. x ,  y >.  e.  _V
21 opex 4554 . . . . . . . . . . . . . . . 16  |-  <. a ,  z >.  e.  _V
22 eleq1 2501 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  <. x ,  y
>.  ->  ( p  e.  F  <->  <. x ,  y
>.  e.  F ) )
2322anbi1d 704 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. x ,  y
>.  ->  ( ( p  e.  F  /\  q  e.  F )  <->  ( <. x ,  y >.  e.  F  /\  q  e.  F
) ) )
24 breq2 4294 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. x ,  y
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) p  <-> 
q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <. x ,  y >. )
)
2523, 24anbi12d 710 . . . . . . . . . . . . . . . 16  |-  ( p  =  <. x ,  y
>.  ->  ( ( ( p  e.  F  /\  q  e.  F )  /\  q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p )  <-> 
( ( <. x ,  y >.  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) <.
x ,  y >.
) ) )
26 eleq1 2501 . . . . . . . . . . . . . . . . . . 19  |-  ( q  =  <. a ,  z
>.  ->  ( q  e.  F  <->  <. a ,  z
>.  e.  F ) )
2726anbi2d 703 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  <. a ,  z
>.  ->  ( ( <.
x ,  y >.  e.  F  /\  q  e.  F )  <->  ( <. x ,  y >.  e.  F  /\  <. a ,  z
>.  e.  F ) ) )
28 breq1 4293 . . . . . . . . . . . . . . . . . . 19  |-  ( q  =  <. a ,  z
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) <.
x ,  y >.  <->  <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.
) )
29 vex 2973 . . . . . . . . . . . . . . . . . . . . 21  |-  x  e. 
_V
30 vex 2973 . . . . . . . . . . . . . . . . . . . . 21  |-  y  e. 
_V
3121, 29, 30brtxp 27909 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y ) )
32 vex 2973 . . . . . . . . . . . . . . . . . . . . . . 23  |-  a  e. 
_V
33 vex 2973 . . . . . . . . . . . . . . . . . . . . . . 23  |-  z  e. 
_V
3432, 33, 29br1steq 27583 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >. 1st x  <->  x  =  a
)
35 equcom 1732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  <->  a  =  x )
3634, 35bitri 249 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >. 1st x  <->  a  =  x )
3721, 30brco 5008 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  E. x
( <. a ,  z
>. 2nd x  /\  x
( _V  \  _I  ) y ) )
3832, 33, 29br2ndeq 27584 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
a ,  z >. 2nd x  <->  x  =  z
)
3938anbi1i 695 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
<. a ,  z >. 2nd x  /\  x
( _V  \  _I  ) y )  <->  ( x  =  z  /\  x
( _V  \  _I  ) y ) )
4039exbii 1634 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( <. a ,  z >. 2nd x  /\  x ( _V  \  _I  ) y )  <->  E. x
( x  =  z  /\  x ( _V 
\  _I  ) y ) )
41 breq1 4293 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  z ( _V  \  _I  ) y ) )
42 brv 27906 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  z _V y
43 brdif 4340 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z ( _V  \  _I  ) y  <->  ( z _V y  /\  -.  z  _I  y ) )
4442, 43mpbiran 909 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z ( _V  \  _I  ) y  <->  -.  z  _I  y )
4530ideq 4990 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  _I  y  <->  z  =  y )
46 equcom 1732 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  =  y  <->  y  =  z )
4745, 46bitri 249 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z  _I  y  <->  y  =  z )
4847notbii 296 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  z  _I  y  <->  -.  y  =  z )
4944, 48bitri 249 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z ( _V  \  _I  ) y  <->  -.  y  =  z )
5041, 49syl6bb 261 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  -.  y  =  z ) )
5133, 50ceqsexv 3007 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( x  =  z  /\  x ( _V  \  _I  )
y )  <->  -.  y  =  z )
5237, 40, 513bitri 271 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  -.  y  =  z )
5336, 52anbi12i 697 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y )  <->  ( a  =  x  /\  -.  y  =  z ) )
5431, 53bitri 249 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  ( a  =  x  /\  -.  y  =  z
) )
5528, 54syl6bb 261 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  <. a ,  z
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) <.
x ,  y >.  <->  ( a  =  x  /\  -.  y  =  z
) ) )
5627, 55anbi12d 710 . . . . . . . . . . . . . . . . 17  |-  ( q  =  <. a ,  z
>.  ->  ( ( (
<. x ,  y >.  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.
)  <->  ( ( <.
x ,  y >.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  ( a  =  x  /\  -.  y  =  z ) ) ) )
57 an12 795 . . . . . . . . . . . . . . . . 17  |-  ( ( ( <. x ,  y
>.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  ( a  =  x  /\  -.  y  =  z ) )  <-> 
( a  =  x  /\  ( ( <.
x ,  y >.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  -.  y  =  z ) ) )
5856, 57syl6bb 261 . . . . . . . . . . . . . . . 16  |-  ( q  =  <. a ,  z
>.  ->  ( ( (
<. x ,  y >.  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.
)  <->  ( a  =  x  /\  ( (
<. x ,  y >.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  -.  y  =  z ) ) ) )
5920, 21, 25, 58ceqsex2v 3009 . . . . . . . . . . . . . . 15  |-  ( E. p E. q ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>.  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  ( a  =  x  /\  (
( <. x ,  y
>.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  -.  y  =  z ) ) )
6019, 59bitr3i 251 . . . . . . . . . . . . . 14  |-  ( E. p E. q ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  ( a  =  x  /\  (
( <. x ,  y
>.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  -.  y  =  z ) ) )
6160exbii 1634 . . . . . . . . . . . . 13  |-  ( E. a E. p E. q ( ( p  =  <. x ,  y
>.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  E. a
( a  =  x  /\  ( ( <.
x ,  y >.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  -.  y  =  z ) ) )
62 opeq1 4057 . . . . . . . . . . . . . . . . 17  |-  ( a  =  x  ->  <. a ,  z >.  =  <. x ,  z >. )
6362eleq1d 2507 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( <. a ,  z >.  e.  F  <->  <. x ,  z
>.  e.  F ) )
6463anbi2d 703 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( <. x ,  y
>.  e.  F  /\  <. a ,  z >.  e.  F
)  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) ) )
6564anbi1d 704 . . . . . . . . . . . . . 14  |-  ( a  =  x  ->  (
( ( <. x ,  y >.  e.  F  /\  <. a ,  z
>.  e.  F )  /\  -.  y  =  z
)  <->  ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  /\  -.  y  =  z ) ) )
6629, 65ceqsexv 3007 . . . . . . . . . . . . 13  |-  ( E. a ( a  =  x  /\  ( (
<. x ,  y >.  e.  F  /\  <. a ,  z >.  e.  F
)  /\  -.  y  =  z ) )  <-> 
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  /\  -.  y  =  z
) )
6761, 66bitri 249 . . . . . . . . . . . 12  |-  ( E. a E. p E. q ( ( p  =  <. x ,  y
>.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  /\  -.  y  =  z ) )
6867exbii 1634 . . . . . . . . . . 11  |-  ( E. z E. a E. p E. q ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  E. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  /\  -.  y  =  z
) )
69 exanali 1637 . . . . . . . . . . 11  |-  ( E. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  /\  -.  y  =  z )  <->  -.  A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )
7068, 69bitri 249 . . . . . . . . . 10  |-  ( E. z E. a E. p E. q ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  -.  A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )
7116, 17, 703bitri 271 . . . . . . . . 9  |-  ( E. p E. q E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  -.  A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )
7271exbii 1634 . . . . . . . 8  |-  ( E. y E. p E. q E. a E. z
( ( p  = 
<. x ,  y >.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  E. y  -.  A. z ( (
<. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) )
73 exnal 1618 . . . . . . . 8  |-  ( E. y  -.  A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z )  <->  -.  A. y A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )
7415, 72, 733bitri 271 . . . . . . 7  |-  ( E. q E. p E. y E. a E. z
( ( p  = 
<. x ,  y >.  /\  q  =  <. a ,  z >. )  /\  ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )  <->  -.  A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) )
7574exbii 1634 . . . . . 6  |-  ( E. x E. q E. p E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  E. x  -.  A. y A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )
76 exnal 1618 . . . . . 6  |-  ( E. x  -.  A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z )  <->  -.  A. x A. y A. z ( ( <. x ,  y
>.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) )
7714, 75, 763bitri 271 . . . . 5  |-  ( E. p E. q E. x E. y E. a E. z ( ( p  =  <. x ,  y >.  /\  q  =  <. a ,  z
>. )  /\  (
( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )  <->  -.  A. x A. y A. z ( ( <. x ,  y
>.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) )
7813, 77syl6bb 261 . . . 4  |-  ( Rel 
F  ->  ( E. p E. q ( ( p  e.  F  /\  q  e.  F )  /\  q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p )  <->  -.  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
7978con2bid 329 . . 3  |-  ( Rel 
F  ->  ( A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z )  <->  -.  E. p E. q
( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
8079pm5.32i 637 . 2  |-  ( ( Rel  F  /\  A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )  <->  ( Rel  F  /\  -.  E. p E. q ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
81 dffun4 5428 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
82 df-funs 27889 . . . 4  |-  Funs  =  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )
8382eleq2i 2505 . . 3  |-  ( F  e.  Funs  <->  F  e.  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
84 eldif 3336 . . 3  |-  ( F  e.  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )  <->  ( F  e.  ~P ( _V  X.  _V )  /\  -.  F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
85 elfuns.1 . . . . . 6  |-  F  e. 
_V
8685elpw 3864 . . . . 5  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  F  C_  ( _V  X.  _V ) )
87 df-rel 4845 . . . . 5  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
8886, 87bitr4i 252 . . . 4  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  Rel  F )
8985elfix 27932 . . . . . 6  |-  ( F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) )  <->  F (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) F )
9085, 85coep 27559 . . . . . . 7  |-  ( F (  _E  o.  (
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) F  <->  E. p  e.  F  F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p )
91 vex 2973 . . . . . . . . 9  |-  p  e. 
_V
9285, 91coepr 27560 . . . . . . . 8  |-  ( F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. q  e.  F  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )
9392rexbii 2738 . . . . . . 7  |-  ( E. p  e.  F  F
( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. p  e.  F  E. q  e.  F  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )
9490, 93bitri 249 . . . . . 6  |-  ( F (  _E  o.  (
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) F  <->  E. p  e.  F  E. q  e.  F  q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p )
95 r2ex 2751 . . . . . 6  |-  ( E. p  e.  F  E. q  e.  F  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p  <->  E. p E. q ( ( p  e.  F  /\  q  e.  F
)  /\  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p ) )
9689, 94, 953bitri 271 . . . . 5  |-  ( F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) )  <->  E. p E. q ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )
9796notbii 296 . . . 4  |-  ( -.  F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) )  <->  -.  E. p E. q ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) )
9888, 97anbi12i 697 . . 3  |-  ( ( F  e.  ~P ( _V  X.  _V )  /\  -.  F  e.  Fix (  _E  o.  (
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )  <-> 
( Rel  F  /\  -.  E. p E. q
( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
9983, 84, 983bitri 271 . 2  |-  ( F  e.  Funs  <->  ( Rel  F  /\  -.  E. p E. q ( ( p  e.  F  /\  q  e.  F )  /\  q
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) p ) ) )
10080, 81, 993bitr4ri 278 1  |-  ( F  e.  Funs  <->  Fun  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2714   _Vcvv 2970    \ cdif 3323    C_ wss 3326   ~Pcpw 3858   <.cop 3881   class class class wbr 4290    _E cep 4628    _I cid 4629    X. cxp 4836   `'ccnv 4837    o. ccom 4842   Rel wrel 4843   Fun wfun 5410   1stc1st 6573   2ndc2nd 6574    (x) ctxp 27858   Fixcfix 27863   Funscfuns 27865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-eprel 4630  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fo 5422  df-fv 5424  df-1st 6575  df-2nd 6576  df-txp 27882  df-fix 27887  df-funs 27889
This theorem is referenced by:  elfunsg  27945  dfrdg4  27979  tfrqfree  27980
  Copyright terms: Public domain W3C validator