Theorem dff2 5799

Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Assertion

Ref	Expression
dff2	|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )

Proof of Theorem dff2

Step	Hyp	Ref	Expression
1		ffn 5489	. . 3 |- ( F : A --> B -> F Fn A )
2		fssxp 5510	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3	1, 2	jca 306	. 2 |- ( F : A --> B -> ( F Fn A /\ F C_ ( A X. B ) ) )
4		rnss 4968	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
5		rnxpss 5175	. . . . 5 |- ran ( A X. B ) C_ B
6	4, 5	sstrdi 3240	. . . 4 |- ( F C_ ( A X. B ) -> ran F C_ B )
7	6	anim2i 342	. . 3 |- ( ( F Fn A /\ F C_ ( A X. B ) ) -> ( F Fn A /\ ran F C_ B ) )
8		df-f 5337	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	7, 8	sylibr 134	. 2 |- ( ( F Fn A /\ F C_ ( A X. B ) ) -> F : A --> B )
10	3, 9	impbii 126	1 |- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   C_ wss 3201   X. cxp 4729   ran crn 4732   Fn wfn 5328   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  mapval2  6890  mpomulf  8229  frecuzrdgtclt  10746  imasaddflemg  13479