Theorem funssxp 4577

Description: Two ways of specifying a partial function from A to B.

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 4451	. . . . . 6 |- (Fun F <-> F Fn dom F)
2	1	biimpi 168	. . . . 5 |- (Fun F -> F Fn dom F)
3		sstr 2625	. . . . . 6 |- ((ran F C_ ran ( A X. B) /\ ran ( A X. B) C_ B) -> ran F C_ B)
4		rnss 4189	. . . . . 6 |- (F C_ (A X. B) -> ran F C_ ran ( A X. B))
5		rnxpss 4344	. . . . . 6 |- ran ( A X. B) C_ B
6	3, 4, 5	sylancl 525	. . . . 5 |- (F C_ (A X. B) -> ran F C_ B)
7	2, 6	anim12i 360	. . . 4 |- ((Fun F /\ F C_ (A X. B)) -> (F Fn dom F /\ ran F C_ B))
8		df-f 4010	. . . 4 |- (F:dom F-->B <-> (F Fn dom F /\ ran F C_ B))
9	7, 8	sylibr 217	. . 3 |- ((Fun F /\ F C_ (A X. B)) -> F:dom F-->B)
10		sstr 2625	. . . . 5 |- ((dom F C_ dom ( A X. B) /\ dom ( A X. B) C_ A) -> dom F C_ A)
11		dmss 4156	. . . . 5 |- (F C_ (A X. B) -> dom F C_ dom ( A X. B))
12		dmxpss 4343	. . . . 5 |- dom ( A X. B) C_ A
13	10, 11, 12	sylancl 525	. . . 4 |- (F C_ (A X. B) -> dom F C_ A)
14	13	adantl 424	. . 3 |- ((Fun F /\ F C_ (A X. B)) -> dom F C_ A)
15	9, 14	jca 310	. 2 |- ((Fun F /\ F C_ (A X. B)) -> (F:dom F-->B /\ dom F C_ A))
16		ffun 4565	. . . 4 |- (F:dom F-->B -> Fun F)
17	16	adantr 425	. . 3 |- ((F:dom F-->B /\ dom F C_ A) -> Fun F)
18		fssxp 4575	. . . 4 |- (F:dom F-->B -> F C_ (dom F X. B))
19		ssid 2634	. . . . 5 |- B C_ B
20		xpss12 4089	. . . . 5 |- ((dom F C_ A /\ B C_ B) -> (dom F X. B) C_ (A X. B))
21	19, 20	mpan2 760	. . . 4 |- (dom F C_ A -> (dom F X. B) C_ (A X. B))
22	18, 21	sylan9ss 2628	. . 3 |- ((F:dom F-->B /\ dom F C_ A) -> F C_ (A X. B))
23	17, 22	jca 310	. 2 |- ((F:dom F-->B /\ dom F C_ A) -> (Fun F /\ F C_ (A X. B)))
24	15, 23	impbii 174	1 |- ((Fun F /\ F C_ (A X. B)) <-> (F:dom F-->B /\ dom F C_ A))

Syntax hints:   <-> wb 163   /\ wa 240   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  elpm2 5396

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

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