Theorem f1oabexg 5425

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 5413	. . . . 5 |- ( f : A -1-1-onto-> B -> f : A --> B )
3	2	anim1i 338	. . . 4 |- ( ( f : A -1-1-onto-> B /\ ph ) -> ( f : A --> B /\ ph ) )
4	3	ss2abi 3200	. . 3 |- { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) }
5		eqid 2157	. . . 4 |- { f | ( f : A --> B /\ ph ) } = { f | ( f : A --> B /\ ph ) }
6	5	fabexg 5356	. . 3 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A --> B /\ ph ) } e. _V )
7		ssexg 4103	. . 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 411	. 2 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9	1, 8	eqeltrid 2244	1 |- ( ( A e. C /\ B e. D ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   {cab 2143   _Vcvv 2712   C_ wss 3102   -->wf 5165   -1-1-onto->wf1o 5168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-rel 4592  df-cnv 4593  df-dm 4595  df-rn 4596  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-f1o 5176

This theorem is referenced by: (None)