Theorem fnopab 5332

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 2528	. 2 |- A. x e. A E! y ph
3		fnopab.2	. . 3 |- F = { <. x , y >. | ( x e. A /\ ph ) }
4	3	fnopabg 5331	. 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 1353   E!weu 2024   e. wcel 2146   A.wral 2453   {copab 4058   Fn wfn 5203

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-fun 5210  df-fn 5211

This theorem is referenced by:  fvopab3g  5581