Theorem dff2 6371

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 6045	. . 3 |- ( F : A --> B -> F Fn A )
2		fssxp 6060	. . 3 |- ( F : A --> B -> F C_ ( A X. B ) )
3	1, 2	jca 554	. 2 |- ( F : A --> B -> ( F Fn A /\ F C_ ( A X. B ) ) )
4		rnss 5354	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
5		rnxpss 5566	. . . . 5 |- ran ( A X. B ) C_ B
6	4, 5	syl6ss 3615	. . . 4 |- ( F C_ ( A X. B ) -> ran F C_ B )
7	6	anim2i 593	. . 3 |- ( ( F Fn A /\ F C_ ( A X. B ) ) -> ( F Fn A /\ ran F C_ B ) )
8		df-f 5892	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	7, 8	sylibr 224	. 2 |- ( ( F Fn A /\ F C_ ( A X. B ) ) -> F : A --> B )
10	3, 9	impbii 199	1 |- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   C_ wss 3574   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  fpr2g  6475  mapval2  7887  cardf2  8769  imasaddflem  16190  imasvscaf  16199