Theorem funssxp 5060

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 4931	. . . . . 6 |- ( Fun F <-> F Fn dom F )
2	1	biimpi 113	. . . . 5 |- ( Fun F -> F Fn dom F )
3		rnss 4564	. . . . . 6 |- ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
4		rnxpss 4754	. . . . . 6 |- ran ( A X. B ) C_ B
5	3, 4	syl6ss 2957	. . . . 5 |- ( F C_ ( A X. B ) -> ran F C_ B )
6	2, 5	anim12i 321	. . . 4 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> ( F Fn dom F /\ ran F C_ B ) )
7		df-f 4906	. . . 4 |- ( F : dom F --> B <-> ( F Fn dom F /\ ran F C_ B ) )
8	6, 7	sylibr 137	. . 3 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> F : dom F --> B )
9		dmss 4534	. . . . 5 |- ( F C_ ( A X. B ) -> dom F C_ dom ( A X. B ) )
10		dmxpss 4753	. . . . 5 |- dom ( A X. B ) C_ A
11	9, 10	syl6ss 2957	. . . 4 |- ( F C_ ( A X. B ) -> dom F C_ A )
12	11	adantl 262	. . 3 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> dom F C_ A )
13	8, 12	jca 290	. 2 |- ( ( Fun F /\ F C_ ( A X. B ) ) -> ( F : dom F --> B /\ dom F C_ A ) )
14		ffun 5048	. . . 4 |- ( F : dom F --> B -> Fun F )
15	14	adantr 261	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> Fun F )
16		fssxp 5058	. . . 4 |- ( F : dom F --> B -> F C_ ( dom F X. B ) )
17		xpss1 4448	. . . 4 |- ( dom F C_ A -> ( dom F X. B ) C_ ( A X. B ) )
18	16, 17	sylan9ss 2958	. . 3 |- ( ( F : dom F --> B /\ dom F C_ A ) -> F C_ ( A X. B ) )
19	15, 18	jca 290	. 2 |- ( ( F : dom F --> B /\ dom F C_ A ) -> ( Fun F /\ F C_ ( A X. B ) ) )
20	13, 19	impbii 117	1 |- ( ( Fun F /\ F C_ ( A X. B ) ) <-> ( F : dom F --> B /\ dom F C_ A ) )

Syntax hints:   /\ wa 97   <-> wb 98   C_ wss 2917   X. cxp 4343   dom cdm 4345   ran crn 4346   Fun wfun 4896   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)