Theorem csbcnvg 4795

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 4043	. . . . 5 |- ( A e. V -> ( [. A / x ]. z F y <-> [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y ) )
2		csbconstg 3063	. . . . . 6 |- ( A e. V -> [_ A / x ]_ z = z )
3		csbconstg 3063	. . . . . 6 |- ( A e. V -> [_ A / x ]_ y = y )
4	2, 3	breq12d 4002	. . . . 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 4055	. . 3 |- ( A e. V -> { <. y , z >. | [. A / x ]. z F y } = { <. y , z >. | z [_ A / x ]_ F y } )
7		csbopabg 4067	. . 3 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | z F y } = { <. y , z >. | [. A / x ]. z F y } )
8		df-cnv 4619	. . . 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 2214	. 2 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ { <. y , z >. | z F y } )
11		df-cnv 4619	. . 3 |- `' F = { <. y , z >. | z F y }
12	11	csbeq2i 3076	. 2 |- [_ A / x ]_ `' F = [_ A / x ]_ { <. y , z >. | z F y }
13	10, 12	eqtr4di 2221	1 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   [.wsbc 2955   [_csb 3049   class class class wbr 3989   {copab 4049   `'ccnv 4610

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619

This theorem is referenced by: (None)