Theorem fssxpOLD 4576

Description: A mapping is a class of ordered pairs.

Assertion

Ref	Expression
fssxpOLD	|- (F:A-->B -> F C_ (A X. B))

Proof of Theorem fssxpOLD

Step	Hyp	Ref	Expression
1		xpss12 4089	. . . 4 |- ((A C_ A /\ ran F C_ B) -> (A X. ran F) C_ (A X. B))
2		ssid 2634	. . . 4 |- A C_ A
3		frn 4569	. . . 4 |- (F:A-->B -> ran F C_ B)
4	1, 2, 3	sylancr 526	. . 3 |- (F:A-->B -> (A X. ran F) C_ (A X. B))
5		fdm 4567	. . . 4 |- (F:A-->B -> dom F = A)
6		xpeq1 4016	. . . 4 |- (dom F = A -> (dom F X. ran F) = (A X. ran F))
7		sseq1 2637	. . . 4 |- ((dom F X. ran F) = (A X. ran F) -> ((dom F X. ran F) C_ (A X. B) <-> (A X. ran F) C_ (A X. B)))
8	5, 6, 7	3syl 24	. . 3 |- (F:A-->B -> ((dom F X. ran F) C_ (A X. B) <-> (A X. ran F) C_ (A X. B)))
9	4, 8	mpbird 213	. 2 |- (F:A-->B -> (dom F X. ran F) C_ (A X. B))
10		frel 4566	. . 3 |- (F:A-->B -> Rel F)
11		relssdmrn 4416	. . 3 |- (Rel F -> F C_ (dom F X. ran F))
12		sstr2 2623	. . 3 |- (F C_ (dom F X. ran F) -> ((dom F X. ran F) C_ (A X. B) -> F C_ (A X. B)))
13	10, 11, 12	3syl 24	. 2 |- (F:A-->B -> ((dom F X. ran F) C_ (A X. B) -> F C_ (A X. B)))
14	9, 13	mpd 29	1 |- (F:A-->B -> F C_ (A X. B))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991  -->wf 3994

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-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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