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Theorem elfuns 24454
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 
b  x  y  z  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 risset 2590 . . . . . . . . 9  |-  ( <.
x ,  z >.  e.  F  <->  E. q  e.  F  q  =  <. x ,  z >. )
2 elrel 4789 . . . . . . . . . . 11  |-  ( ( Rel  F  /\  q  e.  F )  ->  E. a E. b  q  =  <. a ,  b >.
)
3 vex 2791 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
4 vex 2791 . . . . . . . . . . . . . . . 16  |-  z  e. 
_V
53, 4opth 4245 . . . . . . . . . . . . . . 15  |-  ( <.
x ,  z >.  =  <. a ,  b
>. 
<->  ( x  =  a  /\  z  =  b ) )
6 eqcom 2285 . . . . . . . . . . . . . . 15  |-  ( <.
a ,  b >.  =  <. x ,  z
>. 
<-> 
<. x ,  z >.  =  <. a ,  b
>. )
7 vex 2791 . . . . . . . . . . . . . . . . 17  |-  a  e. 
_V
8 vex 2791 . . . . . . . . . . . . . . . . 17  |-  b  e. 
_V
97, 8, 3br1steq 24130 . . . . . . . . . . . . . . . 16  |-  ( <.
a ,  b >. 1st x  <->  x  =  a
)
107, 8, 4br2ndeq 24131 . . . . . . . . . . . . . . . 16  |-  ( <.
a ,  b >. 2nd z  <->  z  =  b )
119, 10anbi12i 678 . . . . . . . . . . . . . . 15  |-  ( (
<. a ,  b >. 1st x  /\  <. a ,  b >. 2nd z
)  <->  ( x  =  a  /\  z  =  b ) )
125, 6, 113bitr4i 268 . . . . . . . . . . . . . 14  |-  ( <.
a ,  b >.  =  <. x ,  z
>. 
<->  ( <. a ,  b
>. 1st x  /\  <. a ,  b >. 2nd z
) )
1312a1i 10 . . . . . . . . . . . . 13  |-  ( q  =  <. a ,  b
>.  ->  ( <. a ,  b >.  =  <. x ,  z >.  <->  ( <. a ,  b >. 1st x  /\  <. a ,  b
>. 2nd z ) ) )
14 eqeq1 2289 . . . . . . . . . . . . 13  |-  ( q  =  <. a ,  b
>.  ->  ( q  = 
<. x ,  z >.  <->  <.
a ,  b >.  =  <. x ,  z
>. ) )
15 breq1 4026 . . . . . . . . . . . . . 14  |-  ( q  =  <. a ,  b
>.  ->  ( q 1st x  <->  <. a ,  b
>. 1st x ) )
16 breq1 4026 . . . . . . . . . . . . . 14  |-  ( q  =  <. a ,  b
>.  ->  ( q 2nd z  <->  <. a ,  b
>. 2nd z ) )
1715, 16anbi12d 691 . . . . . . . . . . . . 13  |-  ( q  =  <. a ,  b
>.  ->  ( ( q 1st x  /\  q 2nd z )  <->  ( <. a ,  b >. 1st x  /\  <. a ,  b
>. 2nd z ) ) )
1813, 14, 173bitr4d 276 . . . . . . . . . . . 12  |-  ( q  =  <. a ,  b
>.  ->  ( q  = 
<. x ,  z >.  <->  ( q 1st x  /\  q 2nd z ) ) )
1918exlimivv 1667 . . . . . . . . . . 11  |-  ( E. a E. b  q  =  <. a ,  b
>.  ->  ( q  = 
<. x ,  z >.  <->  ( q 1st x  /\  q 2nd z ) ) )
202, 19syl 15 . . . . . . . . . 10  |-  ( ( Rel  F  /\  q  e.  F )  ->  (
q  =  <. x ,  z >.  <->  ( q 1st x  /\  q 2nd z ) ) )
2120rexbidva 2560 . . . . . . . . 9  |-  ( Rel 
F  ->  ( E. q  e.  F  q  =  <. x ,  z
>. 
<->  E. q  e.  F  ( q 1st x  /\  q 2nd z ) ) )
221, 21syl5bb 248 . . . . . . . 8  |-  ( Rel 
F  ->  ( <. x ,  z >.  e.  F  <->  E. q  e.  F  ( q 1st x  /\  q 2nd z ) ) )
2322anbi2d 684 . . . . . . 7  |-  ( Rel 
F  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  <->  ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) ) ) )
2423imbi1d 308 . . . . . 6  |-  ( Rel 
F  ->  ( (
( <. x ,  y
>.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z )  <->  ( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z ) ) )
2524albidv 1611 . . . . 5  |-  ( Rel 
F  ->  ( A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z )  <->  A. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  ->  y  =  z ) ) )
26252albidv 1613 . . . 4  |-  ( Rel 
F  ->  ( A. 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  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z ) ) )
2726pm5.32i 618 . . 3  |-  ( ( Rel  F  /\  A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F )  -> 
y  =  z ) )  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z ) ) )
2827bicomi 193 . 2  |-  ( ( Rel  F  /\  A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  ->  y  =  z ) )  <->  ( Rel  F  /\  A. x A. y A. z ( (
<. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
29 eldif 3162 . . 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  ) ) ) )
30 df-funs 24402 . . . 4  |-  Funs  =  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )
3130eleq2i 2347 . . 3  |-  ( F  e.  Funs  <->  F  e.  ( ~P ( _V  X.  _V )  \  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
32 df-rel 4696 . . . . 5  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
33 elfuns.1 . . . . . 6  |-  F  e. 
_V
3433elpw 3631 . . . . 5  |-  ( F  e.  ~P ( _V 
X.  _V )  <->  F  C_  ( _V  X.  _V ) )
3532, 34bitr4i 243 . . . 4  |-  ( Rel 
F  <->  F  e.  ~P ( _V  X.  _V )
)
36 alnex 1530 . . . . 5  |-  ( A. x  -.  E. y E. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
)  <->  -.  E. x E. y E. z ( ( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
37 exanali 1572 . . . . . . . . 9  |-  ( E. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
)  <->  -.  A. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  ->  y  =  z ) )
3837con2bii 322 . . . . . . . 8  |-  ( A. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z )  <->  -.  E. z ( (
<. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
) )
3938albii 1553 . . . . . . 7  |-  ( A. y A. z ( (
<. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z )  <->  A. y  -.  E. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
40 alnex 1530 . . . . . . 7  |-  ( A. y  -.  E. z ( ( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  -.  E. y E. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
) )
4139, 40bitri 240 . . . . . 6  |-  ( A. y A. z ( (
<. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z )  <->  -.  E. y E. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
4241albii 1553 . . . . 5  |-  ( A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  ->  y  =  z )  <->  A. x  -.  E. y E. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
4333elfix 24443 . . . . . . 7  |-  ( 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 )
4433, 33coep 24108 . . . . . . 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 )
45 vex 2791 . . . . . . . . . . 11  |-  p  e. 
_V
4633, 45coepr 24109 . . . . . . . . . 10  |-  ( F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. q  e.  F  q ( 1st  (x)  ( ( _V 
\  _I  )  o. 
2nd ) ) p )
47 vex 2791 . . . . . . . . . . . . 13  |-  q  e. 
_V
4847brtxp2 24421 . . . . . . . . . . . 12  |-  ( q ( 1st  (x)  (
( _V  \  _I  )  o.  2nd )
) p  <->  E. x E. y ( p  = 
<. x ,  y >.  /\  q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )
49 3anass 938 . . . . . . . . . . . . 13  |-  ( ( p  =  <. x ,  y >.  /\  q 1st x  /\  q
( ( _V  \  _I  )  o.  2nd ) y )  <->  ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) ) )
50492exbii 1570 . . . . . . . . . . . 12  |-  ( E. x E. y ( p  =  <. x ,  y >.  /\  q 1st x  /\  q
( ( _V  \  _I  )  o.  2nd ) y )  <->  E. x E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
5148, 50bitri 240 . . . . . . . . . . 11  |-  ( q ( 1st  (x)  (
( _V  \  _I  )  o.  2nd )
) p  <->  E. x E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
5251rexbii 2568 . . . . . . . . . 10  |-  ( E. q  e.  F  q ( 1st  (x)  (
( _V  \  _I  )  o.  2nd )
) p  <->  E. q  e.  F  E. x E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
5346, 52bitri 240 . . . . . . . . 9  |-  ( F ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. q  e.  F  E. x E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
5453rexbii 2568 . . . . . . . 8  |-  ( E. p  e.  F  F
( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. p  e.  F  E. q  e.  F  E. x E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
55 rexcom 2701 . . . . . . . 8  |-  ( E. p  e.  F  E. q  e.  F  E. x E. y ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) )  <->  E. q  e.  F  E. p  e.  F  E. x E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
56 rexcom4 2807 . . . . . . . . . . 11  |-  ( E. p  e.  F  E. x E. y ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) )  <->  E. x E. p  e.  F  E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
57 rexcom4 2807 . . . . . . . . . . . . 13  |-  ( E. p  e.  F  E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. y E. p  e.  F  ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) ) )
58 biidd 228 . . . . . . . . . . . . . . 15  |-  ( p  =  <. x ,  y
>.  ->  ( ( q 1st x  /\  q
( ( _V  \  _I  )  o.  2nd ) y )  <->  ( q 1st x  /\  q
( ( _V  \  _I  )  o.  2nd ) y ) ) )
5958ceqsrexv2 24078 . . . . . . . . . . . . . 14  |-  ( E. p  e.  F  ( p  =  <. x ,  y >.  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
6059exbii 1569 . . . . . . . . . . . . 13  |-  ( E. y E. p  e.  F  ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. y ( <.
x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
6157, 60bitri 240 . . . . . . . . . . . 12  |-  ( E. p  e.  F  E. y ( p  = 
<. x ,  y >.  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. y ( <.
x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
6261exbii 1569 . . . . . . . . . . 11  |-  ( E. x E. p  e.  F  E. y ( p  =  <. x ,  y >.  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. x E. y
( <. x ,  y
>.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
6356, 62bitri 240 . . . . . . . . . 10  |-  ( E. p  e.  F  E. x E. y ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) )  <->  E. x E. y ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
6463rexbii 2568 . . . . . . . . 9  |-  ( E. q  e.  F  E. p  e.  F  E. x E. y ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) )  <->  E. q  e.  F  E. x E. y ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
65 rexcom4 2807 . . . . . . . . 9  |-  ( E. q  e.  F  E. x E. y ( <.
x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. x E. q  e.  F  E. y
( <. x ,  y
>.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
66 rexcom4 2807 . . . . . . . . . . 11  |-  ( E. q  e.  F  E. y ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. y E. q  e.  F  ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) ) )
67 vex 2791 . . . . . . . . . . . . . . . . . . 19  |-  y  e. 
_V
6847, 67brco 4852 . . . . . . . . . . . . . . . . . 18  |-  ( q ( ( _V  \  _I  )  o.  2nd ) y  <->  E. z
( q 2nd z  /\  z ( _V  \  _I  ) y ) )
69 brv 24417 . . . . . . . . . . . . . . . . . . . . . 22  |-  z _V y
70 brdif 4071 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z ( _V  \  _I  ) y  <->  ( z _V y  /\  -.  z  _I  y ) )
7169, 70mpbiran 884 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z ( _V  \  _I  ) y  <->  -.  z  _I  y )
7267ideq 4836 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( z  _I  y  <->  z  =  y )
73 equcom 1647 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  =  z  <->  z  =  y )
7472, 73bitr4i 243 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( z  _I  y  <->  y  =  z )
7574notbii 287 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  z  _I  y  <->  -.  y  =  z )
7671, 75bitri 240 . . . . . . . . . . . . . . . . . . . 20  |-  ( z ( _V  \  _I  ) y  <->  -.  y  =  z )
7776anbi2i 675 . . . . . . . . . . . . . . . . . . 19  |-  ( ( q 2nd z  /\  z ( _V  \  _I  ) y )  <->  ( q 2nd z  /\  -.  y  =  z ) )
7877exbii 1569 . . . . . . . . . . . . . . . . . 18  |-  ( E. z ( q 2nd z  /\  z ( _V  \  _I  )
y )  <->  E. z
( q 2nd z  /\  -.  y  =  z ) )
7968, 78bitri 240 . . . . . . . . . . . . . . . . 17  |-  ( q ( ( _V  \  _I  )  o.  2nd ) y  <->  E. z
( q 2nd z  /\  -.  y  =  z ) )
8079anbi2i 675 . . . . . . . . . . . . . . . 16  |-  ( ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y )  <-> 
( q 1st x  /\  E. z ( q 2nd z  /\  -.  y  =  z )
) )
8180anbi2i 675 . . . . . . . . . . . . . . 15  |-  ( (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  E. z ( q 2nd z  /\  -.  y  =  z )
) ) )
82 19.42v 1846 . . . . . . . . . . . . . . . 16  |-  ( E. z ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  ( q 2nd z  /\  -.  y  =  z ) ) )  <->  ( <. x ,  y >.  e.  F  /\  E. z ( q 1st x  /\  (
q 2nd z  /\  -.  y  =  z
) ) ) )
83 anass 630 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( <. x ,  y
>.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  ( <. x ,  y >.  e.  F  /\  ( ( q 1st x  /\  q 2nd z )  /\  -.  y  =  z )
) )
84 anass 630 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( q 1st x  /\  q 2nd z )  /\  -.  y  =  z )  <->  ( q 1st x  /\  (
q 2nd z  /\  -.  y  =  z
) ) )
8584anbi2i 675 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. x ,  y >.  e.  F  /\  (
( q 1st x  /\  q 2nd z )  /\  -.  y  =  z ) )  <->  ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  ( q 2nd z  /\  -.  y  =  z ) ) ) )
8683, 85bitr2i 241 . . . . . . . . . . . . . . . . 17  |-  ( (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  ( q 2nd z  /\  -.  y  =  z ) ) )  <->  ( ( <. x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
8786exbii 1569 . . . . . . . . . . . . . . . 16  |-  ( E. z ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  ( q 2nd z  /\  -.  y  =  z ) ) )  <->  E. z
( ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
88 19.42v 1846 . . . . . . . . . . . . . . . . 17  |-  ( E. z ( q 1st x  /\  ( q 2nd z  /\  -.  y  =  z )
)  <->  ( q 1st x  /\  E. z
( q 2nd z  /\  -.  y  =  z ) ) )
8988anbi2i 675 . . . . . . . . . . . . . . . 16  |-  ( (
<. x ,  y >.  e.  F  /\  E. z
( q 1st x  /\  ( q 2nd z  /\  -.  y  =  z ) ) )  <->  ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  E. z ( q 2nd z  /\  -.  y  =  z )
) ) )
9082, 87, 893bitr3ri 267 . . . . . . . . . . . . . . 15  |-  ( (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  E. z ( q 2nd z  /\  -.  y  =  z ) ) )  <->  E. z ( (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
9181, 90bitri 240 . . . . . . . . . . . . . 14  |-  ( (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. z ( (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
9291rexbii 2568 . . . . . . . . . . . . 13  |-  ( E. q  e.  F  (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. q  e.  F  E. z ( ( <.
x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
93 rexcom4 2807 . . . . . . . . . . . . 13  |-  ( E. q  e.  F  E. z ( ( <.
x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  E. z E. q  e.  F  ( ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
94 r19.41v 2693 . . . . . . . . . . . . . . 15  |-  ( E. q  e.  F  ( ( <. x ,  y
>.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  ( E. q  e.  F  ( <. x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
95 r19.42v 2694 . . . . . . . . . . . . . . . 16  |-  ( E. q  e.  F  (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  <-> 
( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) ) )
9695anbi1i 676 . . . . . . . . . . . . . . 15  |-  ( ( E. q  e.  F  ( <. x ,  y
>.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  ( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
) )
9794, 96bitri 240 . . . . . . . . . . . . . 14  |-  ( E. q  e.  F  ( ( <. x ,  y
>.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  ( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
) )
9897exbii 1569 . . . . . . . . . . . . 13  |-  ( E. z E. q  e.  F  ( ( <.
x ,  y >.  e.  F  /\  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z )  <->  E. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
9992, 93, 983bitri 262 . . . . . . . . . . . 12  |-  ( E. q  e.  F  (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. z ( (
<. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
) )
10099exbii 1569 . . . . . . . . . . 11  |-  ( E. y E. q  e.  F  ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. y E. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
10166, 100bitri 240 . . . . . . . . . 10  |-  ( E. q  e.  F  E. y ( <. x ,  y >.  e.  F  /\  ( q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. y E. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
102101exbii 1569 . . . . . . . . 9  |-  ( E. x E. q  e.  F  E. y (
<. x ,  y >.  e.  F  /\  (
q 1st x  /\  q ( ( _V 
\  _I  )  o. 
2nd ) y ) )  <->  E. x E. y E. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z
) )
10364, 65, 1023bitri 262 . . . . . . . 8  |-  ( E. q  e.  F  E. p  e.  F  E. x E. y ( p  =  <. x ,  y
>.  /\  ( q 1st x  /\  q ( ( _V  \  _I  )  o.  2nd )
y ) )  <->  E. x E. y E. z ( ( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
10454, 55, 1033bitri 262 . . . . . . 7  |-  ( E. p  e.  F  F
( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) p  <->  E. x E. y E. z ( ( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
10543, 44, 1043bitri 262 . . . . . 6  |-  ( F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) )  <->  E. x E. y E. z ( ( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
106105notbii 287 . . . . 5  |-  ( -.  F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) )  <->  -.  E. x E. y E. z ( ( <. x ,  y
>.  e.  F  /\  E. q  e.  F  (
q 1st x  /\  q 2nd z ) )  /\  -.  y  =  z ) )
10736, 42, 1063bitr4i 268 . . . 4  |-  ( A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  ->  y  =  z )  <->  -.  F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) )
10835, 107anbi12i 678 . . 3  |-  ( ( Rel  F  /\  A. x A. y A. z
( ( <. x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  ->  y  =  z ) )  <->  ( F  e.  ~P ( _V  X.  _V )  /\  -.  F  e.  Fix (  _E  o.  ( ( 1st  (x)  ( ( _V  \  _I  )  o.  2nd ) )  o.  `'  _E  ) ) ) )
10929, 31, 1083bitr4i 268 . 2  |-  ( F  e.  Funs  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  E. q  e.  F  ( q 1st x  /\  q 2nd z ) )  -> 
y  =  z ) ) )
110 dffun4 5267 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. y A. z ( ( <.
x ,  y >.  e.  F  /\  <. x ,  z >.  e.  F
)  ->  y  =  z ) ) )
11128, 109, 1103bitr4i 268 1  |-  ( F  e.  Funs  <->  Fun  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ~Pcpw 3625   <.cop 3643   class class class wbr 4023    _E cep 4303    _I cid 4304    X. cxp 4687   `'ccnv 4688    o. ccom 4693   Rel wrel 4694   Fun wfun 5249   1stc1st 6120   2ndc2nd 6121    (x) ctxp 24373   Fixcfix 24378   Funscfuns 24380
This theorem is referenced by:  elfunsg  24455  dfrdg4  24488  tfrqfree  24489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-txp 24395  df-fix 24400  df-funs 24402
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