Theorem dmopabss 4943

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 4942	. 2 |- dom { <. x , y >. | ( x e. A /\ ph ) } = { x | E. y ( x e. A /\ ph ) }
2		19.42v 1955	. . . 4 |- ( E. y ( x e. A /\ ph ) <-> ( x e. A /\ E. y ph ) )
3	2	abbii 2347	. . 3 |- { x | E. y ( x e. A /\ ph ) } = { x | ( x e. A /\ E. y ph ) }
4		ssab2 3311	. . 3 |- { x | ( x e. A /\ E. y ph ) } C_ A
5	3, 4	eqsstri 3259	. 2 |- { x | E. y ( x e. A /\ ph ) } C_ A
6	1, 5	eqsstri 3259	1 |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A

Syntax hints:   /\ wa 104   E.wex 1540   e. wcel 2202   {cab 2217   C_ wss 3200   {copab 4149   dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-dm 4735

This theorem is referenced by:  opabex  5877