Theorem dmopab3 4969

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

Assertion

Ref	Expression
dmopab3	|- ( A. x e. A E. y ph <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem dmopab3

Step	Hyp	Ref	Expression
1		df-ral 2525	. 2 |- ( A. x e. A E. y ph <-> A. x ( x e. A -> E. y ph ) )
2		pm4.71 389	. . 3 |- ( ( x e. A -> E. y ph ) <-> ( x e. A <-> ( x e. A /\ E. y ph ) ) )
3	2	albii 1519	. 2 |- ( A. x ( x e. A -> E. y ph ) <-> A. x ( x e. A <-> ( x e. A /\ E. y ph ) ) )
4		dmopab 4967	. . . . 5 |- dom { <. x , y >. | ( x e. A /\ ph ) } = { x | E. y ( x e. A /\ ph ) }
5		19.42v 1956	. . . . . 6 |- ( E. y ( x e. A /\ ph ) <-> ( x e. A /\ E. y ph ) )
6	5	abbii 2348	. . . . 5 |- { x | E. y ( x e. A /\ ph ) } = { x | ( x e. A /\ E. y ph ) }
7	4, 6	eqtri 2253	. . . 4 |- dom { <. x , y >. | ( x e. A /\ ph ) } = { x | ( x e. A /\ E. y ph ) }
8	7	eqeq1i 2240	. . 3 |- ( dom { <. x , y >. | ( x e. A /\ ph ) } = A <-> { x | ( x e. A /\ E. y ph ) } = A )
9		eqcom 2234	. . 3 |- ( A = { x | ( x e. A /\ E. y ph ) } <-> { x | ( x e. A /\ E. y ph ) } = A )
10		abeq2 2341	. . 3 |- ( A = { x | ( x e. A /\ E. y ph ) } <-> A. x ( x e. A <-> ( x e. A /\ E. y ph ) ) )
11	8, 9, 10	3bitr2ri 209	. 2 |- ( A. x ( x e. A <-> ( x e. A /\ E. y ph ) ) <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )
12	1, 3, 11	3bitri 206	1 |- ( A. x e. A E. y ph <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   A.wral 2520   {copab 4170   dom cdm 4749

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-dm 4759

This theorem is referenced by:  dmxpm  4977  dmxpid  4978  fnopabg  5482  opabn1stprc  6389  acfun  7514  ccfunen  7578