ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ov Unicode version

Theorem ov 5993
Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ov.1  |-  C  e. 
_V
ov.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ov.3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
ov.4  |-  ( z  =  C  ->  ( ch 
<->  th ) )
ov.5  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
ov.6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
Assertion
Ref Expression
ov  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    x, R, y, z    x, S, y, z    th, x, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    ch( x, y, z)    F( x, y, z)

Proof of Theorem ov
StepHypRef Expression
1 df-ov 5877 . . . . 5  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ov.6 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
32fveq1i 5516 . . . . 5  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
41, 3eqtri 2198 . . . 4  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
54eqeq1i 2185 . . 3  |-  ( ( A F B )  =  C  <->  ( { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )  =  C )
6 ov.5 . . . . . 6  |-  ( ( x  e.  R  /\  y  e.  S )  ->  E! z ph )
76fnoprab 5977 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 eleq1 2240 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
98anbi1d 465 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  y  e.  S ) ) )
10 eleq1 2240 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
1110anbi2d 464 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  B  e.  S ) ) )
129, 11opelopabg 4268 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } 
<->  ( A  e.  R  /\  B  e.  S
) ) )
1312ibir 177 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
14 fnopfvb 5557 . . . . 5  |-  ( ( { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }  /\  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) } )  ->  (
( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )  =  C  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } ) )
157, 13, 14sylancr 414 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) } `
 <. A ,  B >. )  =  C  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } ) )
16 ov.1 . . . . 5  |-  C  e. 
_V
17 ov.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
189, 17anbi12d 473 . . . . . 6  |-  ( x  =  A  ->  (
( ( x  e.  R  /\  y  e.  S )  /\  ph ) 
<->  ( ( A  e.  R  /\  y  e.  S )  /\  ps ) ) )
19 ov.3 . . . . . . 7  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
2011, 19anbi12d 473 . . . . . 6  |-  ( y  =  B  ->  (
( ( A  e.  R  /\  y  e.  S )  /\  ps ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  ch ) ) )
21 ov.4 . . . . . . 7  |-  ( z  =  C  ->  ( ch 
<->  th ) )
2221anbi2d 464 . . . . . 6  |-  ( z  =  C  ->  (
( ( A  e.  R  /\  B  e.  S )  /\  ch ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2318, 20, 22eloprabg 5962 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  <->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2416, 23mp3an3 1326 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }  <->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2515, 24bitrd 188 . . 3  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) } `
 <. A ,  B >. )  =  C  <->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
265, 25bitrid 192 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <-> 
( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2726bianabs 611 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   E!weu 2026    e. wcel 2148   _Vcvv 2737   <.cop 3595   {copab 4063    Fn wfn 5211   ` cfv 5216  (class class class)co 5874   {coprab 5875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-oprab 5878
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator