Theorem fnopabg 5419

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 2131	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
2	1	albii 1494	. . . . 5 |- ( A. x E* y ( x e. A /\ ph ) <-> A. x ( x e. A -> E* y ph ) )
3		funopab 5325	. . . . 5 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } <-> A. x E* y ( x e. A /\ ph ) )
4		df-ral 2491	. . . . 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 4910	. . . 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 2634	. . 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 5293	. . 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 2103	. . . 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 2514	. 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 5387	. 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 1371   = wceq 1373   E.wex 1516   E!weu 2055   E*wmo 2056   e. wcel 2178   A.wral 2486   {copab 4120   dom cdm 4693   Fun wfun 5284   Fn wfn 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-fun 5292  df-fn 5293

This theorem is referenced by:  fnopab  5420  mptfng  5421  uchoice  6246