Theorem fnopabg 5042

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 2016	. . . . . 6 |- ( E* y ( x e. A /\ ph ) <-> ( x e. A -> E* y ph ) )
2	1	albii 1399	. . . . 5 |- ( A. x E* y ( x e. A /\ ph ) <-> A. x ( x e. A -> E* y ph ) )
3		funopab 4955	. . . . 5 |- ( Fun { <. x , y >. | ( x e. A /\ ph ) } <-> A. x E* y ( x e. A /\ ph ) )
4		df-ral 2353	. . . . 5 |- ( A. x e. A E* y ph <-> A. x ( x e. A -> E* y ph ) )
5	2, 3, 4	3bitr4ri 211	. . . 4 |- ( A. x e. A E* y ph <-> Fun { <. x , y >. | ( x e. A /\ ph ) } )
6		dmopab3 4566	. . . 4 |- ( A. x e. A E. y ph <-> dom { <. x , y >. | ( x e. A /\ ph ) } = A )
7	5, 6	anbi12i 447	. . 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 2485	. . 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 4925	. . 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 210	. 2 |- ( A. x e. A ( E* y ph /\ E. y ph ) <-> { <. x , y >. | ( x e. A /\ ph ) } Fn A )
11		eu5 1988	. . . 4 |- ( E! y ph <-> ( E. y ph /\ E* y ph ) )
12		ancom 262	. . . 4 |- ( ( E. y ph /\ E* y ph ) <-> ( E* y ph /\ E. y ph ) )
13	11, 12	bitri 182	. . 3 |- ( E! y ph <-> ( E* y ph /\ E. y ph ) )
14	13	ralbii 2372	. 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 5013	. 2 |- ( F Fn A <-> { <. x , y >. | ( x e. A /\ ph ) } Fn A )
17	10, 14, 16	3bitr4i 210	1 |- ( A. x e. A E! y ph <-> F Fn A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   E*wmo 1942   A.wral 2348   {copab 3838   dom cdm 4363   Fun wfun 4916   Fn wfn 4917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-fun 4924  df-fn 4925

This theorem is referenced by:  fnopab  5043  mptfng  5044