Theorem dvhvscacbv 36387

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 6190	. . . 4 |- ( s = t -> ( s ` ( 1st ` f ) ) = ( t ` ( 1st ` f ) ) )
3		coeq1 5279	. . . 4 |- ( s = t -> ( s o. ( 2nd ` f ) ) = ( t o. ( 2nd ` f ) ) )
4	2, 3	opeq12d 4410	. . 3 |- ( s = t -> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. = <. ( t ` ( 1st ` f ) ) , ( t o. ( 2nd ` f ) ) >. )
5		fveq2 6191	. . . . 5 |- ( f = g -> ( 1st ` f ) = ( 1st ` g ) )
6	5	fveq2d 6195	. . . 4 |- ( f = g -> ( t ` ( 1st ` f ) ) = ( t ` ( 1st ` g ) ) )
7		fveq2 6191	. . . . 5 |- ( f = g -> ( 2nd ` f ) = ( 2nd ` g ) )
8	7	coeq2d 5284	. . . 4 |- ( f = g -> ( t o. ( 2nd ` f ) ) = ( t o. ( 2nd ` g ) ) )
9	6, 8	opeq12d 4410	. . 3 |- ( f = g -> <. ( t ` ( 1st ` f ) ) , ( t o. ( 2nd ` f ) ) >. = <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )
10	4, 9	cbvmpt2v 6735	. 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 2644	1 |- .x. = ( t e. E , g e. ( T X. E ) |-> <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )

Syntax hints:   = wceq 1483   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-co 5123  df-iota 5851  df-fv 5896  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  dvhvscaval  36388