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Theorem elfuns 25761
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 4979 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  p  e.  F )  ->  E. x E. y  p  =  <. x ,  y >.
)
21ex 425 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( p  e.  F  ->  E. x E. y  p  =  <. x ,  y >.
) )
3 elrel 4979 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  q  e.  F )  ->  E. a E. z  q  =  <. a ,  z >.
)
43ex 425 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( q  e.  F  ->  E. a E. z  q  =  <. a ,  z >.
) )
52, 4anim12d 548 . . . . . . . . 9  |-  ( Rel 
F  ->  ( (
p  e.  F  /\  q  e.  F )  ->  ( E. x E. y  p  =  <. x ,  y >.  /\  E. a E. z  q  = 
<. a ,  z >.
) ) )
65adantrd 456 . . . . . . . 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 618 . . . . . . 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 1928 . . . . . . . 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 1941 . . . . . . . . 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 678 . . . . . . . 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 243 . . . . . . 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 254 . . . . . 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 1639 . . . . 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 1759 . . . . . 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 1759 . . . . . . . 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 1761 . . . . . . . . . 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 1757 . . . . . . . . . 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 939 . . . . . . . . . . . . . . . 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 1594 . . . . . . . . . . . . . . 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 4428 . . . . . . . . . . . . . . . 16  |-  <. x ,  y >.  e.  _V
21 opex 4428 . . . . . . . . . . . . . . . 16  |-  <. a ,  z >.  e.  _V
22 eleq1 2497 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  <. x ,  y
>.  ->  ( p  e.  F  <->  <. x ,  y
>.  e.  F ) )
2322anbi1d 687 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. x ,  y
>.  ->  ( ( p  e.  F  /\  q  e.  F )  <->  ( <. x ,  y >.  e.  F  /\  q  e.  F
) ) )
24 breq2 4217 . . . . . . . . . . . . . . . . 17  |-  ( p  =  <. x ,  y
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) p  <-> 
q ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <. x ,  y >. )
)
2523, 24anbi12d 693 . . . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . . . . . . . 19  |-  ( q  =  <. a ,  z
>.  ->  ( q  e.  F  <->  <. a ,  z
>.  e.  F ) )
2726anbi2d 686 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  <. a ,  z
>.  ->  ( ( <.
x ,  y >.  e.  F  /\  q  e.  F )  <->  ( <. x ,  y >.  e.  F  /\  <. a ,  z
>.  e.  F ) ) )
28 breq1 4216 . . . . . . . . . . . . . . . . . . 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 2960 . . . . . . . . . . . . . . . . . . . . 21  |-  x  e. 
_V
30 vex 2960 . . . . . . . . . . . . . . . . . . . . 21  |-  y  e. 
_V
3121, 29, 30brtxp 25726 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y ) )
32 vex 2960 . . . . . . . . . . . . . . . . . . . . . . 23  |-  a  e. 
_V
33 vex 2960 . . . . . . . . . . . . . . . . . . . . . . 23  |-  z  e. 
_V
3432, 33, 29br1steq 25399 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >. 1st x  <->  x  =  a
)
35 equcom 1693 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  <->  a  =  x )
3634, 35bitri 242 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >. 1st x  <->  a  =  x )
3721, 30brco 5044 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  E. x
( <. a ,  z
>. 2nd x  /\  x
( _V  \  _I  ) y ) )
3832, 33, 29br2ndeq 25400 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
a ,  z >. 2nd x  <->  x  =  z
)
3938anbi1i 678 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
<. a ,  z >. 2nd x  /\  x
( _V  \  _I  ) y )  <->  ( x  =  z  /\  x
( _V  \  _I  ) y ) )
4039exbii 1593 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( <. a ,  z >. 2nd x  /\  x ( _V  \  _I  ) y )  <->  E. x
( x  =  z  /\  x ( _V 
\  _I  ) y ) )
41 breq1 4216 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  z ( _V  \  _I  ) y ) )
42 brv 25723 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  z _V y
43 brdif 4261 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z ( _V  \  _I  ) y  <->  ( z _V y  /\  -.  z  _I  y ) )
4442, 43mpbiran 886 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z ( _V  \  _I  ) y  <->  -.  z  _I  y )
4530ideq 5026 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  _I  y  <->  z  =  y )
46 equcom 1693 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( z  =  y  <->  y  =  z )
4745, 46bitri 242 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( z  _I  y  <->  y  =  z )
4847notbii 289 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -.  z  _I  y  <->  -.  y  =  z )
4944, 48bitri 242 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z ( _V  \  _I  ) y  <->  -.  y  =  z )
5041, 49syl6bb 254 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  z  ->  (
x ( _V  \  _I  ) y  <->  -.  y  =  z ) )
5133, 50ceqsexv 2992 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. x ( x  =  z  /\  x ( _V  \  _I  )
y )  <->  -.  y  =  z )
5237, 40, 513bitri 264 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
a ,  z >.
( ( _V  \  _I  )  o.  2nd ) y  <->  -.  y  =  z )
5336, 52anbi12i 680 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. a ,  z >. 1st x  /\  <. a ,  z >. (
( _V  \  _I  )  o.  2nd )
y )  <->  ( a  =  x  /\  -.  y  =  z ) )
5431, 53bitri 242 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
a ,  z >.
( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) ) <.
x ,  y >.  <->  ( a  =  x  /\  -.  y  =  z
) )
5528, 54syl6bb 254 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  <. a ,  z
>.  ->  ( q ( 1st  (x)  ( ( _V  \  _I  )  o. 
2nd ) ) <.
x ,  y >.  <->  ( a  =  x  /\  -.  y  =  z
) ) )
5627, 55anbi12d 693 . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . 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 254 . . . . . . . . . . . . . . . 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 2994 . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . 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 1593 . . . . . . . . . . . . 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 3985 . . . . . . . . . . . . . . . . 17  |-  ( a  =  x  ->  <. a ,  z >.  =  <. x ,  z >. )
6362eleq1d 2503 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  ( <. a ,  z >.  e.  F  <->  <. x ,  z
>.  e.  F ) )
6463anbi2d 686 . . . . . . . . . . . . . . 15  |-  ( a  =  x  ->  (
( <. x ,  y
>.  e.  F  /\  <. a ,  z >.  e.  F
)  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) ) )
6564anbi1d 687 . . . . . . . . . . . . . 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 2992 . . . . . . . . . . . . 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 242 . . . . . . . . . . . 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 1593 . . . . . . . . . . 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 1596 . . . . . . . . . . 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 242 . . . . . . . . . 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 264 . . . . . . . . 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 1593 . . . . . . . 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 1584 . . . . . . . 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 264 . . . . . . 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 1593 . . . . . 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 1584 . . . . . 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 264 . . . . 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 254 . . . 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 321 . . 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 620 . 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 5467 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
82 df-funs 25706 . . . 4  |-  Funs  =  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )
8382eleq2i 2501 . . 3  |-  ( F  e.  Funs  <->  F  e.  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
84 eldif 3331 . . 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 3806 . . . . 5  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  F  C_  ( _V  X.  _V ) )
87 df-rel 4886 . . . . 5  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
8886, 87bitr4i 245 . . . 4  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  Rel  F )
8985elfix 25749 . . . . . 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 25375 . . . . . . 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 2960 . . . . . . . . 9  |-  p  e. 
_V
9285, 91coepr 25376 . . . . . . . 8  |-  ( F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. q  e.  F  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )
9392rexbii 2731 . . . . . . 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 242 . . . . . 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 2744 . . . . . 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 264 . . . . 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 289 . . . 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 680 . . 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 264 . 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 271 1  |-  ( F  e.  Funs  <->  Fun  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2707   _Vcvv 2957    \ cdif 3318    C_ wss 3321   ~Pcpw 3800   <.cop 3818   class class class wbr 4213    _E cep 4493    _I cid 4494    X. cxp 4877   `'ccnv 4878    o. ccom 4883   Rel wrel 4884   Fun wfun 5449   1stc1st 6348   2ndc2nd 6349    (x) ctxp 25675   Fixcfix 25680   Funscfuns 25682
This theorem is referenced by:  elfunsg  25762  dfrdg4  25796  tfrqfree  25797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-eprel 4495  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463  df-1st 6350  df-2nd 6351  df-txp 25699  df-fix 25704  df-funs 25706
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