Theorem funssxp 5300

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 5161	. . . . . 6 |- ( Fun F <-> F Fn dom F )
2	1	biimpi 119	. . . . 5 |- ( Fun F -> F Fn dom F )
3		rnss 4777	. . . . . 6 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
4		rnxpss 4978	. . . . . 6 |- ran ( A X. B ) C_ B
5	3, 4	sstrdi 3114	. . . . 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 5135	. . . 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 4746	. . . . 5 |- ( F C_ ( A X. B ) -> dom F C_ dom ( A X. B ) )
10		dmxpss 4977	. . . . 5 |- dom ( A X. B ) C_ A
11	9, 10	sstrdi 3114	. . . 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 5283	. . . 4 |- ( F : dom F --> B -> Fun F )
15	14	adantr 274	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> Fun F )
16		fssxp 5298	. . . 4 |- ( F : dom F --> B -> F C_ ( dom F X. B ) )
17		xpss1 4657	. . . 4 |- ( dom F C_ A -> ( dom F X. B ) C_ ( A X. B ) )
18	16, 17	sylan9ss 3115	. . 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 3076   X. cxp 4545   dom cdm 4547   ran crn 4548   Fun wfun 5125   Fn wfn 5126   -->wf 5127

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-f 5135

This theorem is referenced by:  elpm2g  6567  casef  6981