Theorem fnopab 5457

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 2585	. 2 |- A. x e. A E! y ph
3		fnopab.2	. . 3 |- F = { <. x , y >. | ( x e. A /\ ph ) }
4	3	fnopabg 5456	. 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 1397   E!weu 2079   e. wcel 2202   A.wral 2510   {copab 4149   Fn wfn 5321

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-ral 2515  df-rex 2516  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-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-fun 5328  df-fn 5329

This theorem is referenced by:  fvopab3g  5719