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Theorem csbpredg 31727
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
StepHypRef Expression
1 csbin 3799 . . 3  |-  [_ A  /  x ]_ ( D  i^i  ( `' R " { X } ) )  =  ( [_ A  /  x ]_ D  i^i  [_ A  /  x ]_ ( `' R " { X } ) )
2 csbima12 5185 . . . . 5  |-  [_ A  /  x ]_ ( `' R " { X } )  =  (
[_ A  /  x ]_ `' R " [_ A  /  x ]_ { X } )
3 csbcnv 5018 . . . . . . 7  |-  `' [_ A  /  x ]_ R  =  [_ A  /  x ]_ `' R
43imaeq1i 5165 . . . . . 6  |-  ( `'
[_ A  /  x ]_ R " [_ A  /  x ]_ { X } )  =  (
[_ A  /  x ]_ `' R " [_ A  /  x ]_ { X } )
5 csbsng 4030 . . . . . . 7  |-  ( A  e.  V  ->  [_ A  /  x ]_ { X }  =  { [_ A  /  x ]_ X }
)
65imaeq2d 5168 . . . . . 6  |-  ( A  e.  V  ->  ( `' [_ A  /  x ]_ R " [_ A  /  x ]_ { X } )  =  ( `' [_ A  /  x ]_ R " { [_ A  /  x ]_ X } ) )
74, 6syl5eqr 2499 . . . . 5  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ `' R " [_ A  /  x ]_ { X } )  =  ( `' [_ A  /  x ]_ R " { [_ A  /  x ]_ X } ) )
82, 7syl5eq 2497 . . . 4  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( `' R " { X } )  =  ( `' [_ A  /  x ]_ R " { [_ A  /  x ]_ X } ) )
98ineq2d 3634 . . 3  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ D  i^i  [_ A  /  x ]_ ( `' R " { X } ) )  =  ( [_ A  /  x ]_ D  i^i  ( `' [_ A  /  x ]_ R " { [_ A  /  x ]_ X } ) ) )
101, 9syl5eq 2497 . 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 5380 . . 3  |-  Pred ( R ,  D ,  X )  =  ( D  i^i  ( `' R " { X } ) )
1211csbeq2i 3782 . 2  |-  [_ A  /  x ]_ Pred ( R ,  D ,  X )  =  [_ A  /  x ]_ ( D  i^i  ( `' R " { X } ) )
13 df-pred 5380 . 2  |-  Pred ( [_ A  /  x ]_ R ,  [_ A  /  x ]_ D ,  [_ A  /  x ]_ X )  =  (
[_ A  /  x ]_ D  i^i  ( `' [_ A  /  x ]_ R " { [_ A  /  x ]_ X } ) )
1410, 12, 133eqtr4g 2510 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ Pred ( R ,  D ,  X )  =  Pred ( [_ A  /  x ]_ R ,  [_ A  /  x ]_ D ,  [_ A  /  x ]_ X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887   [_csb 3363    i^i cin 3403   {csn 3968   `'ccnv 4833   "cima 4837   Predcpred 5379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380
This theorem is referenced by:  csbwrecsg  31728
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