Theorem fnopabg 5293

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 2081	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
2	1	albii 1450	. . . . 5 |- ( A. x E* y ( x e. A /\ ph ) <-> A. x ( x e. A -> E* y ph ) )
3		funopab 5205	. . . . 5 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } <-> A. x E* y ( x e. A /\ ph ) )
4		df-ral 2440	. . . . 5 |- ( A. x e. A E* y ph <-> A. x ( x e. A -> E* y ph ) )
5	2, 3, 4	3bitr4ri 212	. . . 4 |- ( A. x e. A E* y ph <-> Fun { <. x , y >. | ( x e. A /\ ph ) } )
6		dmopab3 4799	. . . 4 |- ( A. x e. A E. y ph <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )
7	5, 6	anbi12i 456	. . 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 2583	. . 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 5173	. . 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 211	. 2 |- ( A. x e. A ( E* y ph /\ E. y ph ) <-> { <. x , y >. | ( x e. A /\ ph ) } Fn A )
11		eu5 2053	. . . 4 |- ( E! y ph <-> ( E. y ph /\ E* y ph ) )
12		ancom 264	. . . 4 |- ( ( E. y ph /\ E* y ph ) <-> ( E* y ph /\ E. y ph ) )
13	11, 12	bitri 183	. . 3 |- ( E! y ph <-> ( E* y ph /\ E. y ph ) )
14	13	ralbii 2463	. 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 5264	. 2 |- ( F Fn A <-> { <. x , y >. | ( x e. A /\ ph ) } Fn A )
17	10, 14, 16	3bitr4i 211	1 |- ( A. x e. A E! y ph <-> F Fn A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   E.wex 1472   E!weu 2006   E*wmo 2007   e. wcel 2128   A.wral 2435   {copab 4024   dom cdm 4586   Fun wfun 5164   Fn wfn 5165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-fun 5172  df-fn 5173

This theorem is referenced by:  fnopab  5294  mptfng  5295