Theorem fnopab 5464

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)

Hypotheses

Ref	Expression
fnopab.1	|- ( x e. A -> E! y ph )
fnopab.2	|- F = { <. x , y >. | ( x e. A /\ ph ) }

Assertion

Ref	Expression
fnopab	|- F Fn A

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)   F( x, y)

Proof of Theorem fnopab

Step	Hyp	Ref	Expression
1		fnopab.1	. . 3 |- ( x e. A -> E! y ph )
2	1	rgen 2586	. 2 |- A. x e. A E! y ph
3		fnopab.2	. . 3 |- F = { <. x , y >. | ( x e. A /\ ph ) }
4	3	fnopabg 5463	. 2 |- ( A. x e. A E! y ph <-> F Fn A )
5	2, 4	mpbi 145	1 |- F Fn A

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E!weu 2079   e. wcel 2202   A.wral 2511   {copab 4154   Fn wfn 5328

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-fun 5335  df-fn 5336

This theorem is referenced by:  fvopab3g  5728