Theorem dff2 5703

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 5404	. . 3 |- ( F : A --> B -> F Fn A )
2		fssxp 5422	. . 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 4893	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
5		rnxpss 5098	. . . . 5 |- ran ( A X. B ) C_ B
6	4, 5	sstrdi 3192	. . . 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 5259	. . 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 3154   X. cxp 4658   ran crn 4661   Fn wfn 5250   -->wf 5251

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-fun 5257  df-fn 5258  df-f 5259

This theorem is referenced by:  mapval2  6734  mpomulf  8011  frecuzrdgtclt  10495  imasaddflemg  12902