Theorem dff2 5629

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 5337	. . 3 |- ( F : A --> B -> F Fn A )
2		fssxp 5355	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3	1, 2	jca 304	. 2 |- ( F : A --> B -> ( F Fn A /\ F C_ ( A X. B ) ) )
4		rnss 4834	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
5		rnxpss 5035	. . . . 5 |- ran ( A X. B ) C_ B
6	4, 5	sstrdi 3154	. . . 4 |- ( F C_ ( A X. B ) -> ran F C_ B )
7	6	anim2i 340	. . 3 |- ( ( F Fn A /\ F C_ ( A X. B ) ) -> ( F Fn A /\ ran F C_ B ) )
8		df-f 5192	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	7, 8	sylibr 133	. 2 |- ( ( F Fn A /\ F C_ ( A X. B ) ) -> F : A --> B )
10	3, 9	impbii 125	1 |- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   C_ wss 3116   X. cxp 4602   ran crn 4605   Fn wfn 5183   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  mapval2  6644  frecuzrdgtclt  10356