Theorem fnopabg 5447

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Hypothesis

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

Assertion

Ref	Expression
fnopabg	|- ( A. x e. A E! y ph <-> F Fn A )

Distinct variable group:   x, y, A

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

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		moanimv 2153	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
2	1	albii 1516	. . . . 5 |- ( A. x E* y ( x e. A /\ ph ) <-> A. x ( x e. A -> E* y ph ) )
3		funopab 5353	. . . . 5 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } <-> A. x E* y ( x e. A /\ ph ) )
4		df-ral 2513	. . . . 5 |- ( A. x e. A E* y ph <-> A. x ( x e. A -> E* y ph ) )
5	2, 3, 4	3bitr4ri 213	. . . 4 |- ( A. x e. A E* y ph <-> Fun { <. x , y >. | ( x e. A /\ ph ) } )
6		dmopab3 4936	. . . 4 |- ( A. x e. A E. y ph <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )
7	5, 6	anbi12i 460	. . 3 |- ( ( A. x e. A E* y ph /\ A. x e. A E. y ph ) <-> ( Fun { <. x , y >. | ( x e. A /\ ph ) } /\ dom { <. x , y >. | ( x e. A /\ ph ) } = A ) )
8		r19.26 2657	. . 3 |- ( A. x e. A ( E* y ph /\ E. y ph ) <-> ( A. x e. A E* y ph /\ A. x e. A E. y ph ) )
9		df-fn 5321	. . 3 |- ( { <. x , y >. | ( x e. A /\ ph ) } Fn A <-> ( Fun { <. x , y >. | ( x e. A /\ ph ) } /\ dom { <. x , y >. | ( x e. A /\ ph ) } = A ) )
10	7, 8, 9	3bitr4i 212	. 2 |- ( A. x e. A ( E* y ph /\ E. y ph ) <-> { <. x , y >. | ( x e. A /\ ph ) } Fn A )
11		eu5 2125	. . . 4 |- ( E! y ph <-> ( E. y ph /\ E* y ph ) )
12		ancom 266	. . . 4 |- ( ( E. y ph /\ E* y ph ) <-> ( E* y ph /\ E. y ph ) )
13	11, 12	bitri 184	. . 3 |- ( E! y ph <-> ( E* y ph /\ E. y ph ) )
14	13	ralbii 2536	. 2 |- ( A. x e. A E! y ph <-> A. x e. A ( E* y ph /\ E. y ph ) )
15		fnopabg.1	. . 3 |- F = { <. x , y >. | ( x e. A /\ ph ) }
16	15	fneq1i 5415	. 2 |- ( F Fn A <-> { <. x , y >. | ( x e. A /\ ph ) } Fn A )
17	10, 14, 16	3bitr4i 212	1 |- ( A. x e. A E! y ph <-> F Fn A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   E!weu 2077   E*wmo 2078   e. wcel 2200   A.wral 2508   {copab 4144   dom cdm 4719   Fun wfun 5312   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  fnopab  5448  mptfng  5449  uchoice  6283