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Theorem dvhvscacbv 34737
Description: Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
Hypothesis
Ref Expression
dvhvscaval.s  |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
Assertion
Ref Expression
dvhvscacbv  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
Distinct variable groups:    f, s,
t, g, E    T, s, f, t, g
Allowed substitution hints:    .x. ( t, f, g, s)

Proof of Theorem dvhvscacbv
StepHypRef Expression
1 dvhvscaval.s . 2  |-  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
2 fveq1 5878 . . . 4  |-  ( s  =  t  ->  (
s `  ( 1st `  f ) )  =  ( t `  ( 1st `  f ) ) )
3 coeq1 4997 . . . 4  |-  ( s  =  t  ->  (
s  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f ) ) )
42, 3opeq12d 4166 . . 3  |-  ( s  =  t  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  f
) ) ,  ( t  o.  ( 2nd `  f ) ) >.
)
5 fveq2 5879 . . . . 5  |-  ( f  =  g  ->  ( 1st `  f )  =  ( 1st `  g
) )
65fveq2d 5883 . . . 4  |-  ( f  =  g  ->  (
t `  ( 1st `  f ) )  =  ( t `  ( 1st `  g ) ) )
7 fveq2 5879 . . . . 5  |-  ( f  =  g  ->  ( 2nd `  f )  =  ( 2nd `  g
) )
87coeq2d 5002 . . . 4  |-  ( f  =  g  ->  (
t  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  g ) ) )
96, 8opeq12d 4166 . . 3  |-  ( f  =  g  ->  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  g
) ) ,  ( t  o.  ( 2nd `  g ) ) >.
)
104, 9cbvmpt2v 6390 . 2  |-  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. )  =  (
t  e.  E , 
g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
111, 10eqtri 2493 1  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   <.cop 3965    X. cxp 4837    o. ccom 4843   ` cfv 5589    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811
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-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-rab 2765  df-v 3033  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-co 4848  df-iota 5553  df-fv 5597  df-oprab 6312  df-mpt2 6313
This theorem is referenced by:  dvhvscaval  34738
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