MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prf2fval Unicode version

Theorem prf2fval 13991
Description: Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfval.k  |-  P  =  ( F ⟨,⟩F  G )
prfval.b  |-  B  =  ( Base `  C
)
prfval.h  |-  H  =  (  Hom  `  C
)
prfval.c  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
prfval.d  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
prf1.x  |-  ( ph  ->  X  e.  B )
prf2.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
prf2fval  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
Distinct variable groups:    B, h    h, F    ph, h    h, G    h, X    h, Y    h, H
Allowed substitution hints:    C( h)    D( h)    P( h)    E( h)

Proof of Theorem prf2fval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfval.k . . . 4  |-  P  =  ( F ⟨,⟩F  G )
2 prfval.b . . . 4  |-  B  =  ( Base `  C
)
3 prfval.h . . . 4  |-  H  =  (  Hom  `  C
)
4 prfval.c . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
5 prfval.d . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
61, 2, 3, 4, 5prfval 13989 . . 3  |-  ( ph  ->  P  =  <. (
x  e.  B  |->  <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  B , 
y  e.  B  |->  ( h  e.  ( x H y )  |->  <.
( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >. )
) >. )
7 fvex 5555 . . . . . 6  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2366 . . . . 5  |-  B  e. 
_V
98mptex 5762 . . . 4  |-  ( x  e.  B  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  e.  _V
108, 8mpt2ex 6214 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) )  e. 
_V
119, 10op2ndd 6147 . . 3  |-  ( P  =  <. ( x  e.  B  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) ,  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) >.  ->  ( 2nd `  P
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) )
126, 11syl 15 . 2  |-  ( ph  ->  ( 2nd `  P
)  =  ( x  e.  B ,  y  e.  B  |->  ( h  e.  ( x H y )  |->  <. (
( x ( 2nd `  F ) y ) `
 h ) ,  ( ( x ( 2nd `  G ) y ) `  h
) >. ) ) )
13 simprl 732 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
14 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
1513, 14oveq12d 5892 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
1613, 14oveq12d 5892 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x ( 2nd `  F ) y )  =  ( X ( 2nd `  F ) Y ) )
1716fveq1d 5543 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( x ( 2nd `  F ) y ) `  h
)  =  ( ( X ( 2nd `  F
) Y ) `  h ) )
1813, 14oveq12d 5892 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x ( 2nd `  G ) y )  =  ( X ( 2nd `  G ) Y ) )
1918fveq1d 5543 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( x ( 2nd `  G ) y ) `  h
)  =  ( ( X ( 2nd `  G
) Y ) `  h ) )
2017, 19opeq12d 3820 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. ( ( x ( 2nd `  F ) y ) `  h
) ,  ( ( x ( 2nd `  G
) y ) `  h ) >.  =  <. ( ( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >. )
2115, 20mpteq12dv 4114 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( h  e.  ( x H y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
)  =  ( h  e.  ( X H Y )  |->  <. (
( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >. ) )
22 prf1.x . 2  |-  ( ph  ->  X  e.  B )
23 prf2.y . 2  |-  ( ph  ->  Y  e.  B )
24 ovex 5899 . . . 4  |-  ( X H Y )  e. 
_V
2524mptex 5762 . . 3  |-  ( h  e.  ( X H Y )  |->  <. (
( X ( 2nd `  F ) Y ) `
 h ) ,  ( ( X ( 2nd `  G ) Y ) `  h
) >. )  e.  _V
2625a1i 10 . 2  |-  ( ph  ->  ( h  e.  ( X H Y ) 
|->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
)  e.  _V )
2712, 21, 22, 23, 26ovmpt2d 5991 1  |-  ( ph  ->  ( X ( 2nd `  P ) Y )  =  ( h  e.  ( X H Y )  |->  <. ( ( X ( 2nd `  F
) Y ) `  h ) ,  ( ( X ( 2nd `  G ) Y ) `
 h ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235    Func cfunc 13744   ⟨,⟩F cprf 13961
This theorem is referenced by:  prf2  13992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-ixp 6834  df-func 13748  df-prf 13965
  Copyright terms: Public domain W3C validator