MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ov Structured version   Unicode version

Theorem ov 6321
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 6204 . . . . 5  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 ov.6 . . . . . 6  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) }
32fveq1i 5801 . . . . 5  |-  ( F `
 <. A ,  B >. )  =  ( {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
41, 3eqtri 2483 . . . 4  |-  ( A F B )  =  ( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  R  /\  y  e.  S )  /\  ph ) } `  <. A ,  B >. )
54eqeq1i 2461 . . 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 6304 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R  /\  y  e.  S
)  /\  ph ) }  Fn  { <. x ,  y >.  |  ( x  e.  R  /\  y  e.  S ) }
8 eleq1 2526 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  R  <->  A  e.  R ) )
98anbi1d 704 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  y  e.  S ) ) )
10 eleq1 2526 . . . . . . . 8  |-  ( y  =  B  ->  (
y  e.  S  <->  B  e.  S ) )
1110anbi2d 703 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  R  /\  y  e.  S
)  <->  ( A  e.  R  /\  B  e.  S ) ) )
129, 11opelopabg 4716 . . . . . 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 242 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  R  /\  y  e.  S ) } )
14 fnopfvb 5843 . . . . 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 663 . . . 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 710 . . . . . 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 710 . . . . . 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 703 . . . . . 6  |-  ( z  =  C  ->  (
( ( A  e.  R  /\  B  e.  S )  /\  ch ) 
<->  ( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2318, 20, 22eloprabg 6289 . . . . 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 1304 . . . 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 253 . . 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, 25syl5bb 257 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <-> 
( ( A  e.  R  /\  B  e.  S )  /\  th ) ) )
2726bianabs 875 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( ( A F B )  =  C  <->  th ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E!weu 2262   _Vcvv 3078   <.cop 3992   {copab 4458    Fn wfn 5522   ` cfv 5527  (class class class)co 6201   {coprab 6202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-ov 6204  df-oprab 6205
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator