Theorem dvhvscacbv 34737

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 5878	. . . 4 |- ( s = t -> ( s ` ( 1st ` f ) ) = ( t ` ( 1st ` f ) ) )
3		coeq1 4997	. . . 4 |- ( s = t -> ( s o. ( 2nd ` f ) ) = ( t o. ( 2nd ` f ) ) )
4	2, 3	opeq12d 4166	. . 3 |- ( s = t -> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. = <. ( t ` ( 1st ` f ) ) , ( t o. ( 2nd ` f ) ) >. )
5		fveq2 5879	. . . . 5 |- ( f = g -> ( 1st ` f ) = ( 1st ` g ) )
6	5	fveq2d 5883	. . . 4 |- ( f = g -> ( t ` ( 1st ` f ) ) = ( t ` ( 1st ` g ) ) )
7		fveq2 5879	. . . . 5 |- ( f = g -> ( 2nd ` f ) = ( 2nd ` g ) )
8	7	coeq2d 5002	. . . 4 |- ( f = g -> ( t o. ( 2nd ` f ) ) = ( t o. ( 2nd ` g ) ) )
9	6, 8	opeq12d 4166	. . 3 |- ( f = g -> <. ( t ` ( 1st ` f ) ) , ( t o. ( 2nd ` f ) ) >. = <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )
10	4, 9	cbvmpt2v 6390	. 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 2493	1 |- .x. = ( t e. E , g e. ( T X. E ) |-> <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )

Syntax hints:   = wceq 1452   <.cop 3965   X. cxp 4837   o. ccom 4843   ` cfv 5589   |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-co 4848  df-iota 5553  df-fv 5597  df-oprab 6312  df-mpt2 6313

This theorem is referenced by:  dvhvscaval  34738