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Theorem dvhvscacbv 31288
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 5524 . . . 4  |-  ( s  =  t  ->  (
s `  ( 1st `  f ) )  =  ( t `  ( 1st `  f ) ) )
3 coeq1 4841 . . . 4  |-  ( s  =  t  ->  (
s  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f ) ) )
42, 3opeq12d 3804 . . 3  |-  ( s  =  t  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  f
) ) ,  ( t  o.  ( 2nd `  f ) ) >.
)
5 fveq2 5525 . . . . 5  |-  ( f  =  g  ->  ( 1st `  f )  =  ( 1st `  g
) )
65fveq2d 5529 . . . 4  |-  ( f  =  g  ->  (
t `  ( 1st `  f ) )  =  ( t `  ( 1st `  g ) ) )
7 fveq2 5525 . . . . 5  |-  ( f  =  g  ->  ( 2nd `  f )  =  ( 2nd `  g
) )
87coeq2d 4846 . . . 4  |-  ( f  =  g  ->  (
t  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  g ) ) )
96, 8opeq12d 3804 . . 3  |-  ( f  =  g  ->  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  g
) ) ,  ( t  o.  ( 2nd `  g ) ) >.
)
104, 9cbvmpt2v 5926 . 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 2303 1  |-  .x.  =  ( t  e.  E ,  g  e.  ( T  X.  E )  |->  <.
( t `  ( 1st `  g ) ) ,  ( t  o.  ( 2nd `  g
) ) >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   <.cop 3643    X. cxp 4687    o. ccom 4693   ` cfv 5255    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  dvhvscaval  31289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-co 4698  df-iota 5219  df-fv 5263  df-oprab 5862  df-mpt2 5863
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