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Theorem elfuns 30732
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 4959 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  p  e.  F )  ->  E. x E. y  p  =  <. x ,  y >.
)
21ex 440 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( p  e.  F  ->  E. x E. y  p  =  <. x ,  y >.
) )
3 elrel 4959 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  q  e.  F )  ->  E. a E. z  q  =  <. a ,  z >.
)
43ex 440 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( q  e.  F  ->  E. a E. z  q  =  <. a ,  z >.
) )
52, 4anim12d 570 . . . . . . . . 9  |-  ( Rel 
F  ->  ( (
p  e.  F  /\  q  e.  F )  ->  ( E. x E. y  p  =  <. x ,  y >.  /\  E. a E. z  q  = 
<. a ,  z >.
) ) )
65adantrd 474 . . . . . . . 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 645 . . . . . . 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 1844 . . . . . . . 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 2091 . . . . . . . . 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 706 . . . . . . . 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 258 . . . . . . 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 269 . . . . . 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 1781 . . . . 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 1941 . . . . . 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 1941 . . . . . . . 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 1943 . . . . . . . . . 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 1938 . . . . . . . . . 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 993 . . . . . . . . . . . . . . . 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 1730 . . . . . . . . . . . . . . 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 4681 . . . . . . . . . . . . . . . 16  |-  <. x ,  y >.  e.  _V
21 opex 4681 . . . . . . . . . . . . . . . 16  |-  <. a ,  z >.  e.  _V
22 eleq1 2528 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  <. x ,  y
>.  ->  ( p  e.  F  <->  <. x ,  y
>.  e.  F ) )
2322anbi1d 716 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. x ,  y
>.  ->  ( ( p  e.  F  /\  q  e.  F )  <->  ( <. x ,  y >.  e.  F  /\  q  e.  F
) ) )
24 breq2 4422 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. x ,  y
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) p  <-> 
q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <. x ,  y >. )
)
2523, 24anbi12d 722 . . . . . . . . . . . . . . . 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 2528 . . . . . . . . . . . . . . . . . . 19  |-  ( q  =  <. a ,  z
>.  ->  ( q  e.  F  <->  <. a ,  z
>.  e.  F ) )
2726anbi2d 715 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  <. a ,  z
>.  ->  ( ( <.
x ,  y >.  e.  F  /\  q  e.  F )  <->  ( <. x ,  y >.  e.  F  /\  <. a ,  z
>.  e.  F ) ) )
28 breq1 4421 . . . . . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . . . . . . . . . 21  |-  x  e. 
_V
30 vex 3060 . . . . . . . . . . . . . . . . . . . . 21  |-  y  e. 
_V
3121, 29, 30brtxp 30697 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y ) )
32 vex 3060 . . . . . . . . . . . . . . . . . . . . . . 23  |-  a  e. 
_V
33 vex 3060 . . . . . . . . . . . . . . . . . . . . . . 23  |-  z  e. 
_V
3432, 33, 29br1steq 30464 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >. 1st x  <->  x  =  a
)
35 equcom 1873 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  <->  a  =  x )
3634, 35bitri 257 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >. 1st x  <->  a  =  x )
3721, 30brco 5027 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  E. x
( <. a ,  z
>. 2nd x  /\  x
( _V  \  _I  ) y ) )
3832, 33, 29br2ndeq 30465 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
a ,  z >. 2nd x  <->  x  =  z
)
3938anbi1i 706 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
<. a ,  z >. 2nd x  /\  x
( _V  \  _I  ) y )  <->  ( x  =  z  /\  x
( _V  \  _I  ) y ) )
4039exbii 1729 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( <. a ,  z >. 2nd x  /\  x ( _V  \  _I  ) y )  <->  E. x
( x  =  z  /\  x ( _V 
\  _I  ) y ) )
41 breq1 4421 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  z ( _V  \  _I  ) y ) )
42 brv 30694 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  z _V y
43 brdif 4469 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z ( _V  \  _I  ) y  <->  ( z _V y  /\  -.  z  _I  y ) )
4442, 43mpbiran 934 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z ( _V  \  _I  ) y  <->  -.  z  _I  y )
4530ideq 5009 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  _I  y  <->  z  =  y )
46 equcom 1873 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  =  y  <->  y  =  z )
4745, 46bitri 257 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z  _I  y  <->  y  =  z )
4847notbii 302 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  z  _I  y  <->  -.  y  =  z )
4944, 48bitri 257 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z ( _V  \  _I  ) y  <->  -.  y  =  z )
5041, 49syl6bb 269 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  -.  y  =  z ) )
5133, 50ceqsexv 3096 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( x  =  z  /\  x ( _V  \  _I  )
y )  <->  -.  y  =  z )
5237, 40, 513bitri 279 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  -.  y  =  z )
5336, 52anbi12i 708 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y )  <->  ( a  =  x  /\  -.  y  =  z ) )
5431, 53bitri 257 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  ( a  =  x  /\  -.  y  =  z
) )
5528, 54syl6bb 269 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  <. a ,  z
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) <.
x ,  y >.  <->  ( a  =  x  /\  -.  y  =  z
) ) )
5627, 55anbi12d 722 . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . 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 269 . . . . . . . . . . . . . . . 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 3099 . . . . . . . . . . . . . . 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 259 . . . . . . . . . . . . . 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 1729 . . . . . . . . . . . . 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 4180 . . . . . . . . . . . . . . . . 17  |-  ( a  =  x  ->  <. a ,  z >.  =  <. x ,  z >. )
6362eleq1d 2524 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( <. a ,  z >.  e.  F  <->  <. x ,  z
>.  e.  F ) )
6463anbi2d 715 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( <. x ,  y
>.  e.  F  /\  <. a ,  z >.  e.  F
)  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) ) )
6564anbi1d 716 . . . . . . . . . . . . . 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 3096 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 1729 . . . . . . . . . . 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 1732 . . . . . . . . . . 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 257 . . . . . . . . . 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 279 . . . . . . . . 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 1729 . . . . . . . 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 1710 . . . . . . . 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 279 . . . . . . 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 1729 . . . . . 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 1710 . . . . . 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 279 . . . . 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 269 . . . 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 335 . . 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 647 . 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 5617 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
82 df-funs 30677 . . . 4  |-  Funs  =  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )
8382eleq2i 2532 . . 3  |-  ( F  e.  Funs  <->  F  e.  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
84 eldif 3426 . . 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 3969 . . . . 5  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  F  C_  ( _V  X.  _V ) )
87 df-rel 4863 . . . . 5  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
8886, 87bitr4i 260 . . . 4  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  Rel  F )
8985elfix 30720 . . . . . 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 30441 . . . . . . 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 3060 . . . . . . . . 9  |-  p  e. 
_V
9285, 91coepr 30442 . . . . . . . 8  |-  ( F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. q  e.  F  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )
9392rexbii 2901 . . . . . . 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 257 . . . . . 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 2925 . . . . . 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 279 . . . . 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 302 . . . 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 708 . . 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 279 . 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 286 1  |-  ( F  e.  Funs  <->  Fun  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991   A.wal 1453    = wceq 1455   E.wex 1674    e. wcel 1898   E.wrex 2750   _Vcvv 3057    \ cdif 3413    C_ wss 3416   ~Pcpw 3963   <.cop 3986   class class class wbr 4418    _E cep 4765    _I cid 4766    X. cxp 4854   `'ccnv 4855    o. ccom 4860   Rel wrel 4861   Fun wfun 5599   1stc1st 6823   2ndc2nd 6824    (x) ctxp 30646   Fixcfix 30651   Funscfuns 30653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-eprel 4767  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fo 5611  df-fv 5613  df-1st 6825  df-2nd 6826  df-txp 30670  df-fix 30675  df-funs 30677
This theorem is referenced by:  elfunsg  30733  dfrecs2  30767  dfrdg4  30768
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