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Theorem dvhvscacbv 31896
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 5727 . . . 4  |-  ( s  =  t  ->  (
s `  ( 1st `  f ) )  =  ( t `  ( 1st `  f ) ) )
3 coeq1 5030 . . . 4  |-  ( s  =  t  ->  (
s  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f ) ) )
42, 3opeq12d 3992 . . 3  |-  ( s  =  t  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  f
) ) ,  ( t  o.  ( 2nd `  f ) ) >.
)
5 fveq2 5728 . . . . 5  |-  ( f  =  g  ->  ( 1st `  f )  =  ( 1st `  g
) )
65fveq2d 5732 . . . 4  |-  ( f  =  g  ->  (
t `  ( 1st `  f ) )  =  ( t `  ( 1st `  g ) ) )
7 fveq2 5728 . . . . 5  |-  ( f  =  g  ->  ( 2nd `  f )  =  ( 2nd `  g
) )
87coeq2d 5035 . . . 4  |-  ( f  =  g  ->  (
t  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  g ) ) )
96, 8opeq12d 3992 . . 3  |-  ( f  =  g  ->  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  g
) ) ,  ( t  o.  ( 2nd `  g ) ) >.
)
104, 9cbvmpt2v 6152 . 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 2456 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 1652   <.cop 3817    X. cxp 4876    o. ccom 4882   ` cfv 5454    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348
This theorem is referenced by:  dvhvscaval  31897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-co 4887  df-iota 5418  df-fv 5462  df-oprab 6085  df-mpt2 6086
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