Theorem csbmpt12 5010

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 5008	. . 3 |- [_ A / x ]_ { <. y , z >. | ( y e. Y /\ z = Z ) } = { <. y , z >. | [. A / x ]. ( y e. Y /\ z = Z ) }
2		sbcan 3478	. . . . 5 |- ( [. A / x ]. ( y e. Y /\ z = Z ) <-> ( [. A / x ]. y e. Y /\ [. A / x ]. z = Z ) )
3		sbcel12 3983	. . . . . . 7 |- ( [. A / x ]. y e. Y <-> [_ A / x ]_ y e. [_ A / x ]_ Y )
4		csbconstg 3546	. . . . . . . 8 |- ( A e. V -> [_ A / x ]_ y = y )
5	4	eleq1d 2686	. . . . . . 7 |- ( A e. V -> ( [_ A / x ]_ y e. [_ A / x ]_ Y <-> y e. [_ A / x ]_ Y ) )
6	3, 5	syl5bb 272	. . . . . 6 |- ( A e. V -> ( [. A / x ]. y e. Y <-> y e. [_ A / x ]_ Y ) )
7		sbceq2g 3990	. . . . . 6 |- ( A e. V -> ( [. A / x ]. z = Z <-> z = [_ A / x ]_ Z ) )
8	6, 7	anbi12d 747	. . . . 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 272	. . . 4 |- ( A e. V -> ( [. A / x ]. ( y e. Y /\ z = Z ) <-> ( y e. [_ A / x ]_ Y /\ z = [_ A / x ]_ Z ) ) )
10	9	opabbidv 4716	. . 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 2668	. 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 4730	. . 3 |- ( y e. Y |-> Z ) = { <. y , z >. | ( y e. Y /\ z = Z ) }
13	12	csbeq2i 3993	. 2 |- [_ A / x ]_ ( y e. Y |-> Z ) = [_ A / x ]_ { <. y , z >. | ( y e. Y /\ z = Z ) }
14		df-mpt 4730	. 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 2681	1 |- ( A e. V -> [_ A / x ]_ ( y e. Y |-> Z ) = ( y e. [_ A / x ]_ Y |-> [_ A / x ]_ Z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533   {copab 4712   |-> cmpt 4729

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-opab 4713  df-mpt 4730

This theorem is referenced by:  csbmpt2  5011  esum2dlem  30154