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Theorem csbmpt22g 31802
Description: Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbmpt22g  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y ,  z  e.  Z  |->  D )  =  ( y  e. 
[_ A  /  x ]_ Y ,  z  e. 
[_ A  /  x ]_ Z  |->  [_ A  /  x ]_ D ) )
Distinct variable groups:    y, A    z, A    y, V    z, V    x, y    x, z
Allowed substitution hints:    A( x)    D( x, y, z)    V( x)    Y( x, y, z)    Z( x, y, z)

Proof of Theorem csbmpt22g
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 csboprabg 31801 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. <.
y ,  z >. ,  d >.  |  ( ( y  e.  Y  /\  z  e.  Z
)  /\  d  =  D ) }  =  { <. <. y ,  z
>. ,  d >.  | 
[. A  /  x ]. ( ( y  e.  Y  /\  z  e.  Z )  /\  d  =  D ) } )
2 sbcan 3298 . . . . 5  |-  ( [. A  /  x ]. (
( y  e.  Y  /\  z  e.  Z
)  /\  d  =  D )  <->  ( [. A  /  x ]. (
y  e.  Y  /\  z  e.  Z )  /\  [. A  /  x ]. d  =  D
) )
3 sbcan 3298 . . . . . . 7  |-  ( [. A  /  x ]. (
y  e.  Y  /\  z  e.  Z )  <->  (
[. A  /  x ]. y  e.  Y  /\  [. A  /  x ]. z  e.  Z
) )
4 sbcel12 3776 . . . . . . . . 9  |-  ( [. A  /  x ]. y  e.  Y  <->  [_ A  /  x ]_ y  e.  [_ A  /  x ]_ Y )
5 csbconstg 3362 . . . . . . . . . 10  |-  ( A  e.  V  ->  [_ A  /  x ]_ y  =  y )
65eleq1d 2533 . . . . . . . . 9  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ y  e.  [_ A  /  x ]_ Y  <->  y  e.  [_ A  /  x ]_ Y ) )
74, 6syl5bb 265 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  Y  <->  y  e.  [_ A  /  x ]_ Y ) )
8 sbcel12 3776 . . . . . . . . 9  |-  ( [. A  /  x ]. z  e.  Z  <->  [_ A  /  x ]_ z  e.  [_ A  /  x ]_ Z )
9 csbconstg 3362 . . . . . . . . . 10  |-  ( A  e.  V  ->  [_ A  /  x ]_ z  =  z )
109eleq1d 2533 . . . . . . . . 9  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ z  e.  [_ A  /  x ]_ Z  <->  z  e.  [_ A  /  x ]_ Z ) )
118, 10syl5bb 265 . . . . . . . 8  |-  ( A  e.  V  ->  ( [. A  /  x ]. z  e.  Z  <->  z  e.  [_ A  /  x ]_ Z ) )
127, 11anbi12d 725 . . . . . . 7  |-  ( A  e.  V  ->  (
( [. A  /  x ]. y  e.  Y  /\  [. A  /  x ]. z  e.  Z
)  <->  ( y  e. 
[_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z ) ) )
133, 12syl5bb 265 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( y  e.  Y  /\  z  e.  Z
)  <->  ( y  e. 
[_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z ) ) )
14 sbceq2g 3783 . . . . . 6  |-  ( A  e.  V  ->  ( [. A  /  x ]. d  =  D  <->  d  =  [_ A  /  x ]_ D ) )
1513, 14anbi12d 725 . . . . 5  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ( y  e.  Y  /\  z  e.  Z
)  /\  [. A  /  x ]. d  =  D )  <->  ( ( y  e.  [_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z )  /\  d  =  [_ A  /  x ]_ D ) ) )
162, 15syl5bb 265 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ( y  e.  Y  /\  z  e.  Z )  /\  d  =  D )  <->  ( (
y  e.  [_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z )  /\  d  =  [_ A  /  x ]_ D ) ) )
1716oprabbidv 6364 . . 3  |-  ( A  e.  V  ->  { <. <.
y ,  z >. ,  d >.  |  [. A  /  x ]. (
( y  e.  Y  /\  z  e.  Z
)  /\  d  =  D ) }  =  { <. <. y ,  z
>. ,  d >.  |  ( ( y  e. 
[_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z )  /\  d  =  [_ A  /  x ]_ D ) } )
181, 17eqtrd 2505 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ { <. <.
y ,  z >. ,  d >.  |  ( ( y  e.  Y  /\  z  e.  Z
)  /\  d  =  D ) }  =  { <. <. y ,  z
>. ,  d >.  |  ( ( y  e. 
[_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z )  /\  d  =  [_ A  /  x ]_ D ) } )
19 df-mpt2 6313 . . 3  |-  ( y  e.  Y ,  z  e.  Z  |->  D )  =  { <. <. y ,  z >. ,  d
>.  |  ( (
y  e.  Y  /\  z  e.  Z )  /\  d  =  D
) }
2019csbeq2i 3786 . 2  |-  [_ A  /  x ]_ ( y  e.  Y ,  z  e.  Z  |->  D )  =  [_ A  /  x ]_ { <. <. y ,  z >. ,  d
>.  |  ( (
y  e.  Y  /\  z  e.  Z )  /\  d  =  D
) }
21 df-mpt2 6313 . 2  |-  ( y  e.  [_ A  /  x ]_ Y ,  z  e.  [_ A  /  x ]_ Z  |->  [_ A  /  x ]_ D )  =  { <. <. y ,  z >. ,  d
>.  |  ( (
y  e.  [_ A  /  x ]_ Y  /\  z  e.  [_ A  /  x ]_ Z )  /\  d  =  [_ A  /  x ]_ D ) }
2218, 20, 213eqtr4g 2530 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( y  e.  Y ,  z  e.  Z  |->  D )  =  ( y  e. 
[_ A  /  x ]_ Y ,  z  e. 
[_ A  /  x ]_ Z  |->  [_ A  /  x ]_ D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   [.wsbc 3255   [_csb 3349   {coprab 6309    |-> cmpt2 6310
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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-in 3397  df-ss 3404  df-nul 3723  df-oprab 6312  df-mpt2 6313
This theorem is referenced by:  csbfinxpg  31850
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