Theorem dff2 5640

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 5347	. . 3 |- ( F : A --> B -> F Fn A )
2		fssxp 5365	. . 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 4841	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
5		rnxpss 5042	. . . . 5 |- ran ( A X. B ) C_ B
6	4, 5	sstrdi 3159	. . . 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 5202	. . 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 3121   X. cxp 4609   ran crn 4612   Fn wfn 5193   -->wf 5194

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  mapval2  6656  frecuzrdgtclt  10377