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Theorem fvproj 28733
Description: Value of a function on pairs, given two projections  F and  G. (Contributed by Thierry Arnoux, 30-Dec-2019.)
Hypotheses
Ref Expression
fvproj.h  |-  H  =  ( x  e.  A ,  y  e.  B  |-> 
<. ( F `  x
) ,  ( G `
 y ) >.
)
fvproj.x  |-  ( ph  ->  X  e.  A )
fvproj.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
fvproj  |-  ( ph  ->  ( H `  <. X ,  Y >. )  =  <. ( F `  X ) ,  ( G `  Y )
>. )
Distinct variable groups:    x, A, y    x, B, y    x, F, y    x, G, y
Allowed substitution hints:    ph( x, y)    H( x, y)    X( x, y)    Y( x, y)

Proof of Theorem fvproj
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6311 . 2  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2 fvproj.x . . 3  |-  ( ph  ->  X  e.  A )
3 fvproj.y . . 3  |-  ( ph  ->  Y  e.  B )
4 fveq2 5879 . . . . 5  |-  ( a  =  X  ->  ( F `  a )  =  ( F `  X ) )
54opeq1d 4164 . . . 4  |-  ( a  =  X  ->  <. ( F `  a ) ,  ( G `  b ) >.  =  <. ( F `  X ) ,  ( G `  b ) >. )
6 fveq2 5879 . . . . 5  |-  ( b  =  Y  ->  ( G `  b )  =  ( G `  Y ) )
76opeq2d 4165 . . . 4  |-  ( b  =  Y  ->  <. ( F `  X ) ,  ( G `  b ) >.  =  <. ( F `  X ) ,  ( G `  Y ) >. )
8 fvproj.h . . . . 5  |-  H  =  ( x  e.  A ,  y  e.  B  |-> 
<. ( F `  x
) ,  ( G `
 y ) >.
)
9 fveq2 5879 . . . . . . 7  |-  ( x  =  a  ->  ( F `  x )  =  ( F `  a ) )
109opeq1d 4164 . . . . . 6  |-  ( x  =  a  ->  <. ( F `  x ) ,  ( G `  y ) >.  =  <. ( F `  a ) ,  ( G `  y ) >. )
11 fveq2 5879 . . . . . . 7  |-  ( y  =  b  ->  ( G `  y )  =  ( G `  b ) )
1211opeq2d 4165 . . . . . 6  |-  ( y  =  b  ->  <. ( F `  a ) ,  ( G `  y ) >.  =  <. ( F `  a ) ,  ( G `  b ) >. )
1310, 12cbvmpt2v 6390 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )  =  ( a  e.  A ,  b  e.  B  |->  <. ( F `  a ) ,  ( G `  b )
>. )
148, 13eqtri 2493 . . . 4  |-  H  =  ( a  e.  A ,  b  e.  B  |-> 
<. ( F `  a
) ,  ( G `
 b ) >.
)
15 opex 4664 . . . 4  |-  <. ( F `  X ) ,  ( G `  Y ) >.  e.  _V
165, 7, 14, 15ovmpt2 6451 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X H Y )  =  <. ( F `  X ) ,  ( G `  Y ) >. )
172, 3, 16syl2anc 673 . 2  |-  ( ph  ->  ( X H Y )  =  <. ( F `  X ) ,  ( G `  Y ) >. )
181, 17syl5eqr 2519 1  |-  ( ph  ->  ( H `  <. X ,  Y >. )  =  <. ( F `  X ) ,  ( G `  Y )
>. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   <.cop 3965   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313
This theorem is referenced by:  fimaproj  28734  qtophaus  28737
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