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

Step	Hyp	Ref	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 ) ) )
4	2, 3	opeq12d 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 ) )
6	5	fveq2d 5852	. . . 4 |- ( f = g -> ( t ` ( 1st ` f ) ) = ( t ` ( 1st ` g ) ) )
7		fveq2 5848	. . . . 5 |- ( f = g -> ( 2nd ` f ) = ( 2nd ` g ) )
8	7	coeq2d 5154	. . . 4 |- ( f = g -> ( t o. ( 2nd ` f ) ) = ( t o. ( 2nd ` g ) ) )
9	6, 8	opeq12d 4211	. . 3 |- ( f = g -> <. ( t ` ( 1st ` f ) ) , ( t o. ( 2nd ` f ) ) >. = <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )
10	4, 9	cbvmpt2v 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 ) ) >. )
11	1, 10	eqtri 2483	1 |- .x. = ( t e. E , g e. ( T X. E ) |-> <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )

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