Theorem fssres 3649

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 3606	. . . . 5 |- ((F Fn A /\ C (_ A) -> (F |` C) Fn C)
2		resss 3389	. . . . . . 7 |- (F |` C) (_ F
3		rnss 3348	. . . . . . 7 |- ((F |` C) (_ F -> ran ( F |` C) (_ ran F)
4	2, 3	ax-mp 7	. . . . . 6 |- ran ( F |` C) (_ ran F
5		sstr 2075	. . . . . 6 |- ((ran ( F |` C) (_ ran F /\ ran F (_ B) -> ran ( F |` C) (_ B)
6	4, 5	mpan 697	. . . . 5 |- (ran F (_ B -> ran ( F |` C) (_ B)
7	1, 6	anim12i 333	. . . 4 |- (((F Fn A /\ C (_ A) /\ ran F (_ B) -> ((F |` C) Fn C /\ ran ( F |` C) (_ B))
8	7	an1rs 491	. . 3 |- (((F Fn A /\ ran F (_ B) /\ C (_ A) -> ((F |` C) Fn C /\ ran ( F |` C) (_ B))
9		df-f 3200	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
10	8, 9	sylanb 451	. 2 |- ((F:A-->B /\ C (_ A) -> ((F |` C) Fn C /\ ran ( F |` C) (_ B))
11		df-f 3200	. 2 |- ((F |` C):C-->B <-> ((F |` C) Fn C /\ ran ( F |` C) (_ B))
12	10, 11	sylibr 200	1 |- ((F:A-->B /\ C (_ A) -> (F |` C):C-->B)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2050  ran crn 3177   |` cres 3178   Fn wfn 3183  -->wf 3184

This theorem is referenced by:  fssres2 3650  mapunen 4508  seq1rn 6323  seqzrn 6558  seq1ublem 6911  rescncf 7272  ruclem13 7523  metreslem 7819  metcnss2 7896  issubgi 8118  ghsubgi 8134  eff1i 8739  effoi 8740  hhssnv 9129

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-fun 3198  df-fn 3199  df-f 3200

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