Theorem oprab4 6726

Description: Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)

Assertion

Ref	Expression
oprab4	|- { <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Distinct variable group:   x, y, z

Allowed substitution hints:   ph( x, y, z)   A( x, y, z)   B( x, y, z)

Proof of Theorem oprab4

Step	Hyp	Ref	Expression
1		opelxp 5146	. . 3 |- ( <. x , y >. e. ( A X. B ) <-> ( x e. A /\ y e. B ) )
2	1	anbi1i 731	. 2 |- ( ( <. x , y >. e. ( A X. B ) /\ ph ) <-> ( ( x e. A /\ y e. B ) /\ ph ) )
3	2	oprabbii 6710	1 |- { <. <. x , y >. , z >. | ( <. x , y >. e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   {coprab 6651

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-xp 5120  df-oprab 6654

This theorem is referenced by: (None)