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.

Step	Hyp	Ref	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 )
6	5	eleq1d 2533	. . . . . . . . 9 |- ( A e. V -> ( [_ A / x ]_ y e. [_ A / x ]_ Y <-> y e. [_ A / x ]_ Y ) )
7	4, 6	syl5bb 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 )
10	9	eleq1d 2533	. . . . . . . . 9 |- ( A e. V -> ( [_ A / x ]_ z e. [_ A / x ]_ Z <-> z e. [_ A / x ]_ Z ) )
11	8, 10	syl5bb 265	. . . . . . . 8 |- ( A e. V -> ( [. A / x ]. z e. Z <-> z e. [_ A / x ]_ Z ) )
12	7, 11	anbi12d 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 ) ) )
13	3, 12	syl5bb 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 ) )
15	13, 14	anbi12d 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 ) ) )
16	2, 15	syl5bb 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 ) ) )
17	16	oprabbidv 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 ) } )
18	1, 17	eqtrd 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 ) }
20	19	csbeq2i 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 ) }
22	18, 20, 21	3eqtr4g 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 ) )

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