Theorem funssxp 5400

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 5261	. . . . . 6 |- ( Fun F <-> F Fn dom F )
2	1	biimpi 120	. . . . 5 |- ( Fun F -> F Fn dom F )
3		rnss 4872	. . . . . 6 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
4		rnxpss 5075	. . . . . 6 |- ran ( A X. B ) C_ B
5	3, 4	sstrdi 3182	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ B )
6	2, 5	anim12i 338	. . . 4 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> ( F Fn dom F /\ ran F C_ B ) )
7		df-f 5235	. . . 4 |- ( F : dom F --> B <-> ( F Fn dom F /\ ran F C_ B ) )
8	6, 7	sylibr 134	. . 3 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> F : dom F --> B )
9		dmss 4841	. . . . 5 |- ( F C_ ( A X. B ) -> dom F C_ dom ( A X. B ) )
10		dmxpss 5074	. . . . 5 |- dom ( A X. B ) C_ A
11	9, 10	sstrdi 3182	. . . 4 |- ( F C_ ( A X. B ) -> dom F C_ A )
12	11	adantl 277	. . 3 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> dom F C_ A )
13	8, 12	jca 306	. 2 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> ( F : dom F --> B /\ dom F C_ A ) )
14		ffun 5383	. . . 4 |- ( F : dom F --> B -> Fun F )
15	14	adantr 276	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> Fun F )
16		fssxp 5398	. . . 4 |- ( F : dom F --> B -> F C_ ( dom F X. B ) )
17		xpss1 4751	. . . 4 |- ( dom F C_ A -> ( dom F X. B ) C_ ( A X. B ) )
18	16, 17	sylan9ss 3183	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> F C_ ( A X. B ) )
19	15, 18	jca 306	. 2 |- ( ( F : dom F --> B /\ dom F C_ A ) -> ( Fun F /\ F C_ ( A X. B ) ) )
20	13, 19	impbii 126	1 |- ( ( Fun F /\ F C_ ( A X. B ) ) <-> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   /\ wa 104   <-> wb 105   C_ wss 3144   X. cxp 4639   dom cdm 4641   ran crn 4642   Fun wfun 5225   Fn wfn 5226   -->wf 5227

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-dm 4651  df-rn 4652  df-fun 5233  df-fn 5234  df-f 5235

This theorem is referenced by:  elpm2g  6683  casef  7105