Theorem fssres 3643

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 3600	. . . . 5 |- ((F Fn A /\ C (_ A) -> (F |` C) Fn C)
2		resss 3383	. . . . . . 7 |- (F |` C) (_ F
3		rnss 3342	. . . . . . 7 |- ((F |` C) (_ F -> ran ( F |` C) (_ ran F)
4	2, 3	ax-mp 7	. . . . . 6 |- ran ( F |` C) (_ ran F
5		sstr 2072	. . . . . 6 |- ((ran ( F |` C) (_ ran F /\ ran F (_ B) -> ran ( F |` C) (_ B)
6	4, 5	mpan 695	. . . . 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 489	. . 3 |- (((F Fn A /\ ran F (_ B) /\ C (_ A) -> ((F |` C) Fn C /\ ran ( F |` C) (_ B))
9		df-f 3194	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
10	8, 9	sylanb 449	. 2 |- ((F:A-->B /\ C (_ A) -> ((F |` C) Fn C /\ ran ( F |` C) (_ B))
11		df-f 3194	. 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 2047  ran crn 3171   |` cres 3172   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  fssres2 3644  mapunen 4502  seq1rn 6322  seqzrn 6557  seq1ublem 6911  rescncf 7272  ruclem13 7522  metreslem 7822  metcnss2 7899  issubgi 8122  ghsubgi 8138  eff1i 8744  effoi 8745  hhssnv 9134

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-fun 3192  df-fn 3193  df-f 3194

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