Theorem fssres 4582

Description: Restriction of a function with a subclass of its domain.

Assertion

Ref	Expression
fssres	|- ((F:A-->B /\ C C_ A) -> (F |` C):C-->B)

Proof of Theorem fssres

Step	Hyp	Ref	Expression
1		fnssres 4526	. . . . 5 |- ((F Fn A /\ C C_ A) -> (F |` C) Fn C)
2		resss 4237	. . . . . . 7 |- (F |` C) C_ F
3		rnss 4189	. . . . . . 7 |- ((F |` C) C_ F -> ran ( F |` C) C_ ran F)
4	2, 3	ax-mp 7	. . . . . 6 |- ran ( F |` C) C_ ran F
5		sstr 2625	. . . . . 6 |- ((ran ( F |` C) C_ ran F /\ ran F C_ B) -> ran ( F |` C) C_ B)
6	4, 5	mpan 759	. . . . 5 |- (ran F C_ B -> ran ( F |` C) C_ B)
7	1, 6	anim12i 360	. . . 4 |- (((F Fn A /\ C C_ A) /\ ran F C_ B) -> ((F |` C) Fn C /\ ran ( F |` C) C_ B))
8	7	an1rs 547	. . 3 |- (((F Fn A /\ ran F C_ B) /\ C C_ A) -> ((F |` C) Fn C /\ ran ( F |` C) C_ B))
9		df-f 4010	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
10	8, 9	sylanb 498	. 2 |- ((F:A-->B /\ C C_ A) -> ((F |` C) Fn C /\ ran ( F |` C) C_ B))
11		df-f 4010	. 2 |- ((F |` C):C-->B <-> ((F |` C) Fn C /\ ran ( F |` C) C_ B))
12	10, 11	sylibr 217	1 |- ((F:A-->B /\ C C_ A) -> (F |` C):C-->B)

Syntax hints:   -> wi 3   /\ wa 240   C_ wss 2593  ran crn 3987   |` cres 3988   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  fssres2 4583  mapunen 5596  seq1rn 7735  seqzrn 7800  seq1ublem 8163  rescncf 8534  ruclem13 8791  metreslem 9099  metcnss2 9177  issubgi 9431  ghsubgi 9446  eff1i 10098  effoi 10099  hhssnv 10767  seqzp2 14716  cnres 15889

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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-fun 4008  df-fn 4009  df-f 4010

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