Theorem dmopabss 5336

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 5335	. 2 |- dom { <. x , y >. | ( x e. A /\ ph ) } = { x | E. y ( x e. A /\ ph ) }
2		19.42v 1918	. . . 4 |- ( E. y ( x e. A /\ ph ) <-> ( x e. A /\ E. y ph ) )
3	2	abbii 2739	. . 3 |- { x | E. y ( x e. A /\ ph ) } = { x | ( x e. A /\ E. y ph ) }
4		ssab2 3686	. . 3 |- { x | ( x e. A /\ E. y ph ) } C_ A
5	3, 4	eqsstri 3635	. 2 |- { x | E. y ( x e. A /\ ph ) } C_ A
6	1, 5	eqsstri 3635	1 |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A

Syntax hints:   /\ wa 384   E.wex 1704   e. wcel 1990   {cab 2608   C_ wss 3574   {copab 4712   dom cdm 5114

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-br 4654  df-opab 4713  df-dm 5124

This theorem is referenced by:  fvopab4ndm  6307  opabex  6483  perpln1  25605  dmadjss  28746  abrexdomjm  29345  abrexdom  33525