Theorem dff1o6 5982

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 5384	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dff13 5974	. . 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 5383	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 464	. 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 1011	. . 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 3302	. . . . . . 7 |- ( ran F = B -> ran F C_ B )
7	6	anim2i 342	. . . . . 6 |- ( ( F Fn A /\ ran F = B ) -> ( F Fn A /\ ran F C_ B ) )
8		df-f 5381	. . . . . 6 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	7, 8	sylibr 134	. . . . 5 |- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
10	9	pm4.71ri 396	. . . 4 |- ( ( F Fn A /\ ran F = B ) <-> ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) )
11	10	anbi1i 462	. . 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 568	. . 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 207	. 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 206	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   A.wral 2528   C_ wss 3220   ran crn 4775   Fn wfn 5372   -->wf 5373   -1-1->wf1 5374   -onto->wfo 5375   -1-1-onto->wf1o 5376   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385

This theorem is used by:  ennnfonelemim  13315