Theorem elaltxp 32082

Description: Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)

Assertion

Ref	Expression
elaltxp	|- ( X e. ( A XX. B ) <-> E. x e. A E. y e. B X = << x , y >> )

Distinct variable groups:   x, A, y   x, B, y   x, X, y

Proof of Theorem elaltxp

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( X e. ( A XX. B ) -> X e. _V )
2		altopex 32067	. . . . 5 |- << x , y >> e. _V
3		eleq1 2689	. . . . 5 |- ( X = << x , y >> -> ( X e. _V <-> << x , y >> e. _V ) )
4	2, 3	mpbiri 248	. . . 4 |- ( X = << x , y >> -> X e. _V )
5	4	a1i 11	. . 3 |- ( ( x e. A /\ y e. B ) -> ( X = << x , y >> -> X e. _V ) )
6	5	rexlimivv 3036	. 2 |- ( E. x e. A E. y e. B X = << x , y >> -> X e. _V )
7		eqeq1 2626	. . . 4 |- ( z = X -> ( z = << x , y >> <-> X = << x , y >> ) )
8	7	2rexbidv 3057	. . 3 |- ( z = X -> ( E. x e. A E. y e. B z = << x , y >> <-> E. x e. A E. y e. B X = << x , y >> ) )
9		df-altxp 32066	. . 3 |- ( A XX. B ) = { z | E. x e. A E. y e. B z = << x , y >> }
10	8, 9	elab2g 3353	. 2 |- ( X e. _V -> ( X e. ( A XX. B ) <-> E. x e. A E. y e. B X = << x , y >> ) )
11	1, 6, 10	pm5.21nii 368	1 |- ( X e. ( A XX. B ) <-> E. x e. A E. y e. B X = << x , y >> )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   <<caltop 32063   XX. caltxp 32064

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-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-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-altop 32065  df-altxp 32066

This theorem is referenced by:  altopelaltxp  32083  altxpsspw  32084