Theorem fnopab 5399

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 2558	. 2 |- A. x e. A E! y ph
3		fnopab.2	. . 3 |- F = { <. x , y >. | ( x e. A /\ ph ) }
4	3	fnopabg 5398	. 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 1372   E!weu 2053   e. wcel 2175   A.wral 2483   {copab 4103   Fn wfn 5265

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-fun 5272  df-fn 5273

This theorem is referenced by:  fvopab3g  5651