Theorem fssres 2764

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

Assertion

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

Proof of Theorem fssres

Step	Hyp	Ref	Expression
1		fnssres 2734	. . . . 5 |- ((F Fn A /\ C (_ A) -> (F |` C) Fn C)
2		resss 2587	. . . . . . 7 |- (F |` C) (_ F
3		rnss 2558	. . . . . . 7 |- ((F |` C) (_ F -> ran (F |` C) (_ ran F)
4	2, 3	ax-mp 6	. . . . . 6 |- ran (F |` C) (_ ran F
5		sstr 1511	. . . . . 6 |- ((ran (F |` C) (_ ran F /\ ran F (_ B) -> ran (F |` C) (_ B)
6	4, 5	mpan 518	. . . . 5 |- (ran F (_ B -> ran (F |` C) (_ B)
7	1, 6	anim12i 268	. . . 4 |- (((F Fn A /\ C (_ A) /\ ran F (_ B) -> ((F |` C) Fn C /\ ran (F |` C) (_ B))
8	7	an1rs 373	. . 3 |- (((F Fn A /\ ran F (_ B) /\ C (_ A) -> ((F |` C) Fn C /\ ran (F |` C) (_ B))
9		df-f 2434	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
10	8, 9	sylanb 344	. 2 |- ((F:A-->B /\ C (_ A) -> ((F |` C) Fn C /\ ran (F |` C) (_ B))
11		df-f 2434	. 2 |- ((F |` C):C-->B <-> ((F |` C) Fn C /\ ran (F |` C) (_ B))
12	10, 11	sylibr 175	1 |- ((F:A-->B /\ C (_ A) -> (F |` C):C-->B)

Syntax hints:   -> wi 2   /\ wa 196   (_ wss 1487  ran crn 2411   |` cres 2412   Fn wfn 2417  -->wf 2418

This theorem is referenced by:  mapunen 3397  seqrn2 4674  ruclem13 4897

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-fun 2432  df-fn 2433  df-f 2434

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