Theorem csbcnvg 4723

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 3982	. . . . 5 |- ( A e. V -> ( [. A / x ]. z F y <-> [_ A / x ]_ z [_ A / x ]_ F [_ A / x ]_ y ) )
2		csbconstg 3016	. . . . . 6 |- ( A e. V -> [_ A / x ]_ z = z )
3		csbconstg 3016	. . . . . 6 |- ( A e. V -> [_ A / x ]_ y = y )
4	2, 3	breq12d 3942	. . . . 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 3994	. . 3 |- ( A e. V -> { <. y , z >. | [. A / x ]. z F y } = { <. y , z >. | z [_ A / x ]_ F y } )
7		csbopabg 4006	. . 3 |- ( A e. V -> [_ A / x ]_ { <. y , z >. | z F y } = { <. y , z >. | [. A / x ]. z F y } )
8		df-cnv 4547	. . . 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 2183	. 2 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ { <. y , z >. | z F y } )
11		df-cnv 4547	. . 3 |- `' F = { <. y , z >. | z F y }
12	11	csbeq2i 3029	. 2 |- [_ A / x ]_ `' F = [_ A / x ]_ { <. y , z >. | z F y }
13	10, 12	syl6eqr 2190	1 |- ( A e. V -> `' [_ A / x ]_ F = [_ A / x ]_ `' F )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   [.wsbc 2909   [_csb 3003   class class class wbr 3929   {copab 3988   `'ccnv 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547

This theorem is referenced by: (None)