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Theorem elfuns 29770
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 5114 . . . . . . . . . . 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 5114 . . . . . . . . . . 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 1775 . . . . . . . 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 1991 . . . . . . . . 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 1717 . . . . 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 1852 . . . . . 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 1852 . . . . . . . 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 1854 . . . . . . . . . 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 1850 . . . . . . . . . 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 975 . . . . . . . . . . . . . . . 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 1669 . . . . . . . . . . . . . . 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 4720 . . . . . . . . . . . . . . . 16  |-  <. x ,  y >.  e.  _V
21 opex 4720 . . . . . . . . . . . . . . . 16  |-  <. a ,  z >.  e.  _V
22 eleq1 2529 . . . . . . . . . . . . . . . . . 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 4460 . . . . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . . . . . . 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 4459 . . . . . . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . . . . . . . 21  |-  x  e. 
_V
30 vex 3112 . . . . . . . . . . . . . . . . . . . . 21  |-  y  e. 
_V
3121, 29, 30brtxp 29735 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y ) )
32 vex 3112 . . . . . . . . . . . . . . . . . . . . . . 23  |-  a  e. 
_V
33 vex 3112 . . . . . . . . . . . . . . . . . . . . . . 23  |-  z  e. 
_V
3432, 33, 29br1steq 29421 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >. 1st x  <->  x  =  a
)
35 equcom 1795 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  <->  a  =  x )
3634, 35bitri 249 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >. 1st x  <->  a  =  x )
3721, 30brco 5183 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  E. x
( <. a ,  z
>. 2nd x  /\  x
( _V  \  _I  ) y ) )
3832, 33, 29br2ndeq 29422 . . . . . . . . . . . . . . . . . . . . . . . 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 1668 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( <. a ,  z >. 2nd x  /\  x ( _V  \  _I  ) y )  <->  E. x
( x  =  z  /\  x ( _V 
\  _I  ) y ) )
41 breq1 4459 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  z ( _V  \  _I  ) y ) )
42 brv 29732 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  z _V y
43 brdif 4506 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z ( _V  \  _I  ) y  <->  ( z _V y  /\  -.  z  _I  y ) )
4442, 43mpbiran 918 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z ( _V  \  _I  ) y  <->  -.  z  _I  y )
4530ideq 5165 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  _I  y  <->  z  =  y )
46 equcom 1795 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . . . . . . . . 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 797 . . . . . . . . . . . . . . . . 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 3148 . . . . . . . . . . . . . . 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 1668 . . . . . . . . . . . . 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 4219 . . . . . . . . . . . . . . . . 17  |-  ( a  =  x  ->  <. a ,  z >.  =  <. x ,  z >. )
6362eleq1d 2526 . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . 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 1668 . . . . . . . . . . 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 1671 . . . . . . . . . . 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 1668 . . . . . . . 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 1649 . . . . . . . 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 1668 . . . . . 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 1649 . . . . . 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 5606 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
82 df-funs 29715 . . . 4  |-  Funs  =  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )
8382eleq2i 2535 . . 3  |-  ( F  e.  Funs  <->  F  e.  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
84 eldif 3481 . . 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 4021 . . . . 5  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  F  C_  ( _V  X.  _V ) )
87 df-rel 5015 . . . . 5  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
8886, 87bitr4i 252 . . . 4  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  Rel  F )
8985elfix 29758 . . . . . 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 29398 . . . . . . 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 3112 . . . . . . . . 9  |-  p  e. 
_V
9285, 91coepr 29399 . . . . . . . 8  |-  ( F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. q  e.  F  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )
9392rexbii 2959 . . . . . . 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 2980 . . . . . 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 973   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   E.wrex 2808   _Vcvv 3109    \ cdif 3468    C_ wss 3471   ~Pcpw 4015   <.cop 4038   class class class wbr 4456    _E cep 4798    _I cid 4799    X. cxp 5006   `'ccnv 5007    o. ccom 5012   Rel wrel 5013   Fun wfun 5588   1stc1st 6797   2ndc2nd 6798    (x) ctxp 29684   Fixcfix 29689   Funscfuns 29691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-eprel 4800  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6799  df-2nd 6800  df-txp 29708  df-fix 29713  df-funs 29715
This theorem is referenced by:  elfunsg  29771  dfrdg4  29805  tfrqfree  29806
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