Theorem csbcnvg 4537

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 3834	. . . . 5 |- ( A e. V -> ( [. A / x ]. z F y <-> [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y ) )
2		csbconstg 2920	. . . . . 6 |- ( A e. V -> [_ A / x ]_ z = z )
3		csbconstg 2920	. . . . . 6 |- ( A e. V -> [_ A / x ]_ y = y )
4	2, 3	breq12d 3798	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y <-> z [_ A / x ]_ F y ) )
5	1, 4	bitrd 186	. . . 4 |- ( A e. V -> ( [. A / x ]. z F y <-> z [_ A / x ]_ F y ) )
6	5	opabbidv 3844	. . 3 |- ( A e. V -> { <. y , z >. | [. A / x ]. z F y } = { <. y , z >. | z [_ A / x ]_ F y } )
7		csbopabg 3856	. . 3 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | z F y } = { <. y , z >. | [. A / x ]. z F y } )
8		df-cnv 4371	. . . 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 2124	. 2 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ { <. y , z >. | z F y } )
11		df-cnv 4371	. . 3 |- `' F = { <. y , z >. | z F y }
12	11	csbeq2i 2932	. 2 |- [_ A / x ]_ `' F = [_ A / x ]_ { <. y , z >. | z F y }
13	10, 12	syl6eqr 2131	1 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   [.wsbc 2815   [_csb 2908   class class class wbr 3785   {copab 3838   `'ccnv 4362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371

This theorem is referenced by: (None)