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.

Step	Hyp	Ref	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 )
5	4	eleq1d 2533	. . . . . . 7 |- ( A e. V -> ( [_ A / x ]_ y e. [_ A / x ]_ Y <-> y e. [_ A / x ]_ Y ) )
6	3, 5	syl5bb 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 ) )
8	6, 7	anbi12d 725	. . . . 5 |- ( A e. V -> ( ( [. A / x ]. y e. Y /\ [. A / x ]. z = Z ) <-> ( y e. [_ A / x ]_ Y /\ z = [_ A / x ]_ Z ) ) )
9	2, 8	syl5bb 265	. . . 4 |- ( A e. V -> ( [. A / x ]. ( y e. Y /\ z = Z ) <-> ( y e. [_ A / x ]_ Y /\ z = [_ A / x ]_ Z ) ) )
10	9	opabbidv 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 ) } )
11	1, 10	syl5eq 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 ) }
13	12	csbeq2i 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 ) }
15	11, 13, 14	3eqtr4g 2530	1 |- ( A e. V -> [_ A / x ]_ ( y e. Y |-> Z ) = ( y e. [_ A / x ]_ Y |-> [_ A / x ]_ Z ) )

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