Theorem f1oabexg 5475

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 5463	. . . . 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 3229	. . 3 |- { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) }
5		eqid 2177	. . . 4 |- { f | ( f : A --> B /\ ph ) } = { f | ( f : A --> B /\ ph ) }
6	5	fabexg 5405	. . 3 |- ( ( A e. C /\ B e. D ) -> { f | ( f : A --> B /\ ph ) } e. _V )
7		ssexg 4144	. . 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 2264	1 |- ( ( A e. C /\ B e. D ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   _Vcvv 2739   C_ wss 3131   -->wf 5214   -1-1-onto->wf1o 5217

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-f1o 5225

This theorem is referenced by: (None)