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Theorem dvhvscacbv 35025
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 5774 . . . 4  |-  ( s  =  t  ->  (
s `  ( 1st `  f ) )  =  ( t `  ( 1st `  f ) ) )
3 coeq1 5081 . . . 4  |-  ( s  =  t  ->  (
s  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  f ) ) )
42, 3opeq12d 4151 . . 3  |-  ( s  =  t  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  f
) ) ,  ( t  o.  ( 2nd `  f ) ) >.
)
5 fveq2 5775 . . . . 5  |-  ( f  =  g  ->  ( 1st `  f )  =  ( 1st `  g
) )
65fveq2d 5779 . . . 4  |-  ( f  =  g  ->  (
t `  ( 1st `  f ) )  =  ( t `  ( 1st `  g ) ) )
7 fveq2 5775 . . . . 5  |-  ( f  =  g  ->  ( 2nd `  f )  =  ( 2nd `  g
) )
87coeq2d 5086 . . . 4  |-  ( f  =  g  ->  (
t  o.  ( 2nd `  f ) )  =  ( t  o.  ( 2nd `  g ) ) )
96, 8opeq12d 4151 . . 3  |-  ( f  =  g  ->  <. (
t `  ( 1st `  f ) ) ,  ( t  o.  ( 2nd `  f ) )
>.  =  <. ( t `
 ( 1st `  g
) ) ,  ( t  o.  ( 2nd `  g ) ) >.
)
104, 9cbvmpt2v 6251 . 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 2478 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 1370   <.cop 3967    X. cxp 4922    o. ccom 4928   ` cfv 5502    |-> cmpt2 6178   1stc1st 6661   2ndc2nd 6662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-co 4933  df-iota 5465  df-fv 5510  df-oprab 6180  df-mpt2 6181
This theorem is referenced by:  dvhvscaval  35026
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