Theorem fnopabg 5381

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 2120	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
2	1	albii 1484	. . . . 5 |- ( A. x E* y ( x e. A /\ ph ) <-> A. x ( x e. A -> E* y ph ) )
3		funopab 5293	. . . . 5 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } <-> A. x E* y ( x e. A /\ ph ) )
4		df-ral 2480	. . . . 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 4879	. . . 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 2623	. . 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 5261	. . 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 2092	. . . 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 2503	. 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 5352	. 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 1362   = wceq 1364   E.wex 1506   E!weu 2045   E*wmo 2046   e. wcel 2167   A.wral 2475   {copab 4093   dom cdm 4663   Fun wfun 5252   Fn wfn 5253

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-fun 5260  df-fn 5261

This theorem is referenced by:  fnopab  5382  mptfng  5383  uchoice  6195