Theorem csbcnvg 4788

Description: Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)

Assertion

Ref	Expression
csbcnvg	|- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Proof of Theorem csbcnvg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcbrg 4036	. . . . 5 |- ( A e. V -> ( [. A / x ]. z F y <-> [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y ) )
2		csbconstg 3059	. . . . . 6 |- ( A e. V -> [_ A / x ]_ z = z )
3		csbconstg 3059	. . . . . 6 |- ( A e. V -> [_ A / x ]_ y = y )
4	2, 3	breq12d 3995	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y <-> z [_ A / x ]_ F y ) )
5	1, 4	bitrd 187	. . . 4 |- ( A e. V -> ( [. A / x ]. z F y <-> z [_ A / x ]_ F y ) )
6	5	opabbidv 4048	. . 3 |- ( A e. V -> { <. y , z >. | [. A / x ]. z F y } = { <. y , z >. | z [_ A / x ]_ F y } )
7		csbopabg 4060	. . 3 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | z F y } = { <. y , z >. | [. A / x ]. z F y } )
8		df-cnv 4612	. . . 4 |- `' [_ A / x ]_ F = { <. y , z >. | z [_ A / x ]_ F y }
9	8	a1i 9	. . 3 |- ( A e. V -> `' [_ A / x ]_ F = { <. y , z >. | z [_ A / x ]_ F y } )
10	6, 7, 9	3eqtr4rd 2209	. 2 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ { <. y , z >. | z F y } )
11		df-cnv 4612	. . 3 |- `' F = { <. y , z >. | z F y }
12	11	csbeq2i 3072	. 2 |- [_ A / x ]_ `' F = [_ A / x ]_ { <. y , z >. | z F y }
13	10, 12	eqtr4di 2217	1 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   [.wsbc 2951   [_csb 3045   class class class wbr 3982   {copab 4042   `'ccnv 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612

This theorem is referenced by: (None)