Theorem dff1o6 5868

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 5297	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dff13 5860	. . 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 5296	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2, 3	anbi12i 460	. 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 983	. . 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 3255	. . . . . . 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 5294	. . . . . 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 392	. . . 4 |- ( ( F Fn A /\ ran F = B ) <-> ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) )
11	10	anbi1i 458	. . 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 562	. . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   A.wral 2486   C_ wss 3174   ran crn 4694   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   -onto->wfo 5288   -1-1-onto->wf1o 5289   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298

This theorem is referenced by:  ennnfonelemim  12910