Theorem bj-mpt2mptALT 33072

Description: Alternate proof of mpt2mpt 6752. (Contributed by BJ, 30-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bj-mpt2mptALT.1	|- ( z = <. x , y >. -> C = D )

Assertion

Ref	Expression
bj-mpt2mptALT	|- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y   z, D

Allowed substitution hints:   C( z)   D( x, y)

Proof of Theorem bj-mpt2mptALT

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp2 5132	. . . . 5 |- ( z e. ( A X. B ) <-> E. x e. A E. y e. B z = <. x , y >. )
2	1	anbi1i 731	. . . 4 |- ( ( z e. ( A X. B ) /\ t = C ) <-> ( E. x e. A E. y e. B z = <. x , y >. /\ t = C ) )
3		r19.41v 3089	. . . 4 |- ( E. x e. A ( E. y e. B z = <. x , y >. /\ t = C ) <-> ( E. x e. A E. y e. B z = <. x , y >. /\ t = C ) )
4		r19.41v 3089	. . . . . 6 |- ( E. y e. B ( z = <. x , y >. /\ t = C ) <-> ( E. y e. B z = <. x , y >. /\ t = C ) )
5		bj-mpt2mptALT.1	. . . . . . . . 9 |- ( z = <. x , y >. -> C = D )
6	5	eqeq2d 2632	. . . . . . . 8 |- ( z = <. x , y >. -> ( t = C <-> t = D ) )
7	6	pm5.32i 669	. . . . . . 7 |- ( ( z = <. x , y >. /\ t = C ) <-> ( z = <. x , y >. /\ t = D ) )
8	7	rexbii 3041	. . . . . 6 |- ( E. y e. B ( z = <. x , y >. /\ t = C ) <-> E. y e. B ( z = <. x , y >. /\ t = D ) )
9	4, 8	bitr3i 266	. . . . 5 |- ( ( E. y e. B z = <. x , y >. /\ t = C ) <-> E. y e. B ( z = <. x , y >. /\ t = D ) )
10	9	rexbii 3041	. . . 4 |- ( E. x e. A ( E. y e. B z = <. x , y >. /\ t = C ) <-> E. x e. A E. y e. B ( z = <. x , y >. /\ t = D ) )
11	2, 3, 10	3bitr2i 288	. . 3 |- ( ( z e. ( A X. B ) /\ t = C ) <-> E. x e. A E. y e. B ( z = <. x , y >. /\ t = D ) )
12	11	opabbii 4717	. 2 |- { <. z , t >. | ( z e. ( A X. B ) /\ t = C ) } = { <. z , t >. | E. x e. A E. y e. B ( z = <. x , y >. /\ t = D ) }
13		df-mpt 4730	. 2 |- ( z e. ( A X. B ) |-> C ) = { <. z , t >. | ( z e. ( A X. B ) /\ t = C ) }
14		bj-dfmpt2a 33071	. 2 |- ( x e. A , y e. B |-> D ) = { <. z , t >. | E. x e. A E. y e. B ( z = <. x , y >. /\ t = D ) }
15	12, 13, 14	3eqtr4i 2654	1 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   {copab 4712   |-> cmpt 4729   X. cxp 5112   |-> cmpt2 6652

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-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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  df-xp 5120  df-oprab 6654  df-mpt2 6655

This theorem is referenced by: (None)