Theorem funssxp 5339

Description: Two ways of specifying a partial function from A to B. (Contributed by NM, 13-Nov-2007.)

Assertion

Ref	Expression
funssxp	|- ( ( Fun F /\ F C_ ( A X. B ) ) <-> ( F : dom F --> B /\ dom F C_ A ) )

Proof of Theorem funssxp

Step	Hyp	Ref	Expression
1		funfn 5200	. . . . . 6 |- ( Fun F <-> F Fn dom F )
2	1	biimpi 119	. . . . 5 |- ( Fun F -> F Fn dom F )
3		rnss 4816	. . . . . 6 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
4		rnxpss 5017	. . . . . 6 |- ran ( A X. B ) C_ B
5	3, 4	sstrdi 3140	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ B )
6	2, 5	anim12i 336	. . . 4 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> ( F Fn dom F /\ ran F C_ B ) )
7		df-f 5174	. . . 4 |- ( F : dom F --> B <-> ( F Fn dom F /\ ran F C_ B ) )
8	6, 7	sylibr 133	. . 3 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> F : dom F --> B )
9		dmss 4785	. . . . 5 |- ( F C_ ( A X. B ) -> dom F C_ dom ( A X. B ) )
10		dmxpss 5016	. . . . 5 |- dom ( A X. B ) C_ A
11	9, 10	sstrdi 3140	. . . 4 |- ( F C_ ( A X. B ) -> dom F C_ A )
12	11	adantl 275	. . 3 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> dom F C_ A )
13	8, 12	jca 304	. 2 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> ( F : dom F --> B /\ dom F C_ A ) )
14		ffun 5322	. . . 4 |- ( F : dom F --> B -> Fun F )
15	14	adantr 274	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> Fun F )
16		fssxp 5337	. . . 4 |- ( F : dom F --> B -> F C_ ( dom F X. B ) )
17		xpss1 4696	. . . 4 |- ( dom F C_ A -> ( dom F X. B ) C_ ( A X. B ) )
18	16, 17	sylan9ss 3141	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> F C_ ( A X. B ) )
19	15, 18	jca 304	. 2 |- ( ( F : dom F --> B /\ dom F C_ A ) -> ( Fun F /\ F C_ ( A X. B ) ) )
20	13, 19	impbii 125	1 |- ( ( Fun F /\ F C_ ( A X. B ) ) <-> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   /\ wa 103   <-> wb 104   C_ wss 3102   X. cxp 4584   dom cdm 4586   ran crn 4587   Fun wfun 5164   Fn wfn 5165   -->wf 5166

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-fun 5172  df-fn 5173  df-f 5174

This theorem is referenced by:  elpm2g  6610  casef  7032