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

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 5774	. . . 4 |- ( s = t -> ( s ` ( 1st ` f ) ) = ( t ` ( 1st ` f ) ) )
3		coeq1 5081	. . . 4 |- ( s = t -> ( s o. ( 2nd ` f ) ) = ( t o. ( 2nd ` f ) ) )
4	2, 3	opeq12d 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 ) )
6	5	fveq2d 5779	. . . 4 |- ( f = g -> ( t ` ( 1st ` f ) ) = ( t ` ( 1st ` g ) ) )
7		fveq2 5775	. . . . 5 |- ( f = g -> ( 2nd ` f ) = ( 2nd ` g ) )
8	7	coeq2d 5086	. . . 4 |- ( f = g -> ( t o. ( 2nd ` f ) ) = ( t o. ( 2nd ` g ) ) )
9	6, 8	opeq12d 4151	. . 3 |- ( f = g -> <. ( t ` ( 1st ` f ) ) , ( t o. ( 2nd ` f ) ) >. = <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )
10	4, 9	cbvmpt2v 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 ) ) >. )
11	1, 10	eqtri 2478	1 |- .x. = ( t e. E , g e. ( T X. E ) |-> <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )

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