Theorem funssxp 5357

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 5218	. . . . . 6 |- ( Fun F <-> F Fn dom F )
2	1	biimpi 119	. . . . 5 |- ( Fun F -> F Fn dom F )
3		rnss 4834	. . . . . 6 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
4		rnxpss 5035	. . . . . 6 |- ran ( A X. B ) C_ B
5	3, 4	sstrdi 3154	. . . . 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 5192	. . . 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 4803	. . . . 5 |- ( F C_ ( A X. B ) -> dom F C_ dom ( A X. B ) )
10		dmxpss 5034	. . . . 5 |- dom ( A X. B ) C_ A
11	9, 10	sstrdi 3154	. . . 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 5340	. . . 4 |- ( F : dom F --> B -> Fun F )
15	14	adantr 274	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> Fun F )
16		fssxp 5355	. . . 4 |- ( F : dom F --> B -> F C_ ( dom F X. B ) )
17		xpss1 4714	. . . 4 |- ( dom F C_ A -> ( dom F X. B ) C_ ( A X. B ) )
18	16, 17	sylan9ss 3155	. . 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 3116   X. cxp 4602   dom cdm 4604   ran crn 4605   Fun wfun 5182   Fn wfn 5183   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  elpm2g  6631  casef  7053