Theorem dmopabss 4934

Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Assertion

Ref	Expression
dmopabss	|- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem dmopabss

Step	Hyp	Ref	Expression
1		dmopab 4933	. 2 |- dom { <. x , y >. | ( x e. A /\ ph ) } = { x | E. y ( x e. A /\ ph ) }
2		19.42v 1953	. . . 4 |- ( E. y ( x e. A /\ ph ) <-> ( x e. A /\ E. y ph ) )
3	2	abbii 2345	. . 3 |- { x | E. y ( x e. A /\ ph ) } = { x | ( x e. A /\ E. y ph ) }
4		ssab2 3308	. . 3 |- { x | ( x e. A /\ E. y ph ) } C_ A
5	3, 4	eqsstri 3256	. 2 |- { x | E. y ( x e. A /\ ph ) } C_ A
6	1, 5	eqsstri 3256	1 |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A

Syntax hints:   /\ wa 104   E.wex 1538   e. wcel 2200   {cab 2215   C_ wss 3197   {copab 4143   dom cdm 4718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-dm 4728

This theorem is referenced by:  opabex  5862