Theorem dff1o6 5416

Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)

Assertion

Ref	Expression
dff1o6	|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )

Distinct variable groups:   x, y, A   x, F, y

Allowed substitution hints:   B( x, y)

Proof of Theorem dff1o6

Step	Hyp	Ref	Expression
1		df-f1o 4909	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dff13 5407	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
3		df-fo 4908	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 433	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) /\ ( F Fn A /\ ran F = B ) ) )
5		df-3an 887	. . 3 |- ( ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( ( F Fn A /\ ran F = B ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
6		eqimss 2997	. . . . . . 7 |- ( ran F = B -> ran F C_ B )
7	6	anim2i 324	. . . . . 6 |- ( ( F Fn A /\ ran F = B ) -> ( F Fn A /\ ran F C_ B ) )
8		df-f 4906	. . . . . 6 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	7, 8	sylibr 137	. . . . 5 |- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
10	9	pm4.71ri 372	. . . 4 |- ( ( F Fn A /\ ran F = B ) <-> ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) )
11	10	anbi1i 431	. . 3 |- ( ( ( F Fn A /\ ran F = B ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
12		an32 496	. . 3 |- ( ( ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) /\ ( F Fn A /\ ran F = B ) ) )
13	5, 11, 12	3bitrri 196	. 2 |- ( ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
14	1, 4, 13	3bitri 195	1 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   A.wral 2306   C_ wss 2917   ran crn 4346   Fn wfn 4897   -->wf 4898   -1-1->wf1 4899   -onto->wfo 4900   -1-1-onto->wf1o 4901   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910

This theorem is referenced by: (None)