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Theorem dvhvscacbv 37222
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 5847 . . . 4  |-  ( s  =  t  ->  (
s `  ( 1st `  f ) )  =  ( t `  ( 1st `  f ) ) )
3 coeq1 5149 . . . 4  |-  ( s  =  t  ->  (
s  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f ) ) )
42, 3opeq12d 4211 . . 3  |-  ( s  =  t  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  f
) ) ,  ( t  o.  ( 2nd `  f ) ) >.
)
5 fveq2 5848 . . . . 5  |-  ( f  =  g  ->  ( 1st `  f )  =  ( 1st `  g
) )
65fveq2d 5852 . . . 4  |-  ( f  =  g  ->  (
t `  ( 1st `  f ) )  =  ( t `  ( 1st `  g ) ) )
7 fveq2 5848 . . . . 5  |-  ( f  =  g  ->  ( 2nd `  f )  =  ( 2nd `  g
) )
87coeq2d 5154 . . . 4  |-  ( f  =  g  ->  (
t  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  g ) ) )
96, 8opeq12d 4211 . . 3  |-  ( f  =  g  ->  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  g
) ) ,  ( t  o.  ( 2nd `  g ) ) >.
)
104, 9cbvmpt2v 6350 . 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 2483 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 1398   <.cop 4022    X. cxp 4986    o. ccom 4992   ` cfv 5570    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-co 4997  df-iota 5534  df-fv 5578  df-oprab 6274  df-mpt2 6275
This theorem is referenced by:  dvhvscaval  37223
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