Theorem xp2 7203

Description: Representation of Cartesian product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
xp2	|- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) }

Distinct variable groups:   x, A   x, B

Proof of Theorem xp2

Step	Hyp	Ref	Expression
1		elxp7 7201	. . 3 |- ( x e. ( A X. B ) <-> ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) )
2	1	abbi2i 2738	. 2 |- ( A X. B ) = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) }
3		df-rab 2921	. 2 |- { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) } = { x | ( x e. ( _V X. _V ) /\ ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) ) }
4	2, 3	eqtr4i 2647	1 |- ( A X. B ) = { x e. ( _V X. _V ) | ( ( 1st ` x ) e. A /\ ( 2nd ` x ) e. B ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  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-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  unielxp  7204  xpinpreima  29952  xpinpreima2  29953