Theorem csbpredg 33172

Description: Move class substitution in and out of the predecessor class of a relationship. (Contributed by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbpredg	|- ( A e. V -> [_ A / x ]_ Pred ( R , D , X ) = Pred ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ X ) )

Proof of Theorem csbpredg

Step	Hyp	Ref	Expression
1		csbin 4010	. . 3 |- [_ A / x ]_ ( D i^i ( `' R " { X } ) ) = ( [_ A / x ]_ D i^i [_ A / x ]_ ( `' R " { X } ) )
2		csbima12 5483	. . . . 5 |- [_ A / x ]_ ( `' R " { X } ) = ( [_ A / x ]_ `' R " [_ A / x ]_ { X } )
3		csbcnv 5306	. . . . . . 7 |- `' [_ A / x ]_ R = [_ A / x ]_ `' R
4	3	imaeq1i 5463	. . . . . 6 |- ( `' [_ A / x ]_ R " [_ A / x ]_ { X } ) = ( [_ A / x ]_ `' R " [_ A / x ]_ { X } )
5		csbsng 4243	. . . . . . 7 |- ( A e. V -> [_ A / x ]_ { X } = { [_ A / x ]_ X } )
6	5	imaeq2d 5466	. . . . . 6 |- ( A e. V -> ( `' [_ A / x ]_ R " [_ A / x ]_ { X } ) = ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) )
7	4, 6	syl5eqr 2670	. . . . 5 |- ( A e. V -> ( [_ A / x ]_ `' R " [_ A / x ]_ { X } ) = ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) )
8	2, 7	syl5eq 2668	. . . 4 |- ( A e. V -> [_ A / x ]_ ( `' R " { X } ) = ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) )
9	8	ineq2d 3814	. . 3 |- ( A e. V -> ( [_ A / x ]_ D i^i [_ A / x ]_ ( `' R " { X } ) ) = ( [_ A / x ]_ D i^i ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) )
10	1, 9	syl5eq 2668	. 2 |- ( A e. V -> [_ A / x ]_ ( D i^i ( `' R " { X } ) ) = ( [_ A / x ]_ D i^i ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) ) )
11		df-pred 5680	. . 3 |- Pred ( R , D , X ) = ( D i^i ( `' R " { X } ) )
12	11	csbeq2i 3993	. 2 |- [_ A / x ]_ Pred ( R , D , X ) = [_ A / x ]_ ( D i^i ( `' R " { X } ) )
13		df-pred 5680	. 2 |- Pred ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ X ) = ( [_ A / x ]_ D i^i ( `' [_ A / x ]_ R " { [_ A / x ]_ X } ) )
14	10, 12, 13	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_ Pred ( R , D , X ) = Pred ( [_ A / x ]_ R , [_ A / x ]_ D , [_ A / x ]_ X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   [_csb 3533   i^i cin 3573   {csn 4177   `'ccnv 5113   "cima 5117   Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  csbwrecsg  33173