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Theorem csbmpt12 4735
Description: Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.)
Assertion
Ref Expression
csbmpt12  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  ( y  e. 
[_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z ) )
Distinct variable groups:    y, A    y, V    y, Y    x, y
Allowed substitution hints:    A( x)    V( x)    Y( x)    Z( x, y)

Proof of Theorem csbmpt12
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbopab 4733 . . 3  |-  [_ A  /  x ]_ { <. y ,  z >.  |  ( y  e.  Y  /\  z  =  Z ) }  =  { <. y ,  z >.  |  [. A  /  x ]. (
y  e.  Y  /\  z  =  Z ) }
2 sbcan 3298 . . . . 5  |-  ( [. A  /  x ]. (
y  e.  Y  /\  z  =  Z )  <->  (
[. A  /  x ]. y  e.  Y  /\  [. A  /  x ]. z  =  Z
) )
3 sbcel12 3776 . . . . . . 7  |-  ( [. A  /  x ]. y  e.  Y  <->  [_ A  /  x ]_ y  e.  [_ A  /  x ]_ Y )
4 csbconstg 3362 . . . . . . . 8  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
54eleq1d 2533 . . . . . . 7  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ y  e.  [_ A  /  x ]_ Y  <->  y  e.  [_ A  /  x ]_ Y ) )
63, 5syl5bb 265 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  Y  <->  y  e.  [_ A  /  x ]_ Y ) )
7 sbceq2g 3783 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  =  Z  <->  z  =  [_ A  /  x ]_ Z ) )
86, 7anbi12d 725 . . . . 5  |-  ( A  e.  V  ->  (
( [. A  /  x ]. y  e.  Y  /\  [. A  /  x ]. z  =  Z
)  <->  ( y  e. 
[_ A  /  x ]_ Y  /\  z  =  [_ A  /  x ]_ Z ) ) )
92, 8syl5bb 265 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  Y  /\  z  =  Z
)  <->  ( y  e. 
[_ A  /  x ]_ Y  /\  z  =  [_ A  /  x ]_ Z ) ) )
109opabbidv 4459 . . 3  |-  ( A  e.  V  ->  { <. y ,  z >.  |  [. A  /  x ]. (
y  e.  Y  /\  z  =  Z ) }  =  { <. y ,  z >.  |  ( y  e.  [_ A  /  x ]_ Y  /\  z  =  [_ A  /  x ]_ Z ) } )
111, 10syl5eq 2517 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. y ,  z >.  |  ( y  e.  Y  /\  z  =  Z ) }  =  { <. y ,  z >.  |  ( y  e.  [_ A  /  x ]_ Y  /\  z  =  [_ A  /  x ]_ Z ) } )
12 df-mpt 4456 . . 3  |-  ( y  e.  Y  |->  Z )  =  { <. y ,  z >.  |  ( y  e.  Y  /\  z  =  Z ) }
1312csbeq2i 3786 . 2  |-  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  [_ A  /  x ]_ { <. y ,  z >.  |  ( y  e.  Y  /\  z  =  Z ) }
14 df-mpt 4456 . 2  |-  ( y  e.  [_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z )  =  { <. y ,  z >.  |  ( y  e.  [_ A  /  x ]_ Y  /\  z  =  [_ A  /  x ]_ Z ) }
1511, 13, 143eqtr4g 2530 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y  |->  Z )  =  ( y  e. 
[_ A  /  x ]_ Y  |->  [_ A  /  x ]_ Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   [.wsbc 3255   [_csb 3349   {copab 4453    |-> cmpt 4454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-mpt 4456
This theorem is referenced by:  csbmpt2  4736  esum2dlem  28987
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