Theorem f1oabexg 5595

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 5583	. . . . 5 |- ( f : A -1-1-onto-> B -> f : A --> B )
3	2	anim1i 340	. . . 4 |- ( ( f : A -1-1-onto-> B /\ ph ) -> ( f : A --> B /\ ph ) )
4	3	ss2abi 3299	. . 3 |- { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) }
5		eqid 2231	. . . 4 |- { f | ( f : A --> B /\ ph ) } = { f | ( f : A --> B /\ ph ) }
6	5	fabexg 5524	. . 3 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A --> B /\ ph ) } e. _V )
7		ssexg 4228	. . 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 414	. 2 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9	1, 8	eqeltrid 2318	1 |- ( ( A e. C /\ B e. D ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   _Vcvv 2802   C_ wss 3200   -->wf 5322   -1-1-onto->wf1o 5325

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-f1o 5333

This theorem is referenced by: (None)