Theorem f1oabexg 5158

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypothesis

Ref	Expression
f1oabexg.1	|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }

Assertion

Ref	Expression
f1oabexg	|- ( ( A e. C /\ B e. D ) -> F e. _V )

Distinct variable groups:   A, f   B, f

Allowed substitution hints:   ph( f)   C( f)   D( f)   F( f)

Proof of Theorem f1oabexg

Step	Hyp	Ref	Expression
1		f1oabexg.1	. 2 |- F = { f | ( f : A -1-1-onto-> B /\ ph ) }
2		f1of 5146	. . . . 5 |- ( f : A -1-1-onto-> B -> f : A --> B )
3	2	anim1i 333	. . . 4 |- ( ( f : A -1-1-onto-> B /\ ph ) -> ( f : A --> B /\ ph ) )
4	3	ss2abi 3066	. . 3 |- { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) }
5		eqid 2081	. . . 4 |- { f | ( f : A --> B /\ ph ) } = { f | ( f : A --> B /\ ph ) }
6	5	fabexg 5097	. . 3 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A --> B /\ ph ) } e. _V )
7		ssexg 3917	. . 3 |- ( ( { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) } /\ { f | ( f : A --> B /\ ph ) } e. _V ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
8	4, 6, 7	sylancr 405	. 2 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9	1, 8	syl5eqel 2165	1 |- ( ( A e. C /\ B e. D ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   _Vcvv 2601   C_ wss 2973   -->wf 4918   -1-1-onto->wf1o 4921

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-13 1444  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  ax-un 4188

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-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-f1o 4929

This theorem is referenced by: (None)