Theorem resfunexg 3579

Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28.

Assertion

Ref	Expression
resfunexg	|- ((Fun A /\ B e. C) -> (A |` B) e. V)

Proof of Theorem resfunexg

Step	Hyp	Ref	Expression
1		dmresexg 3382	. . . 4 |- (B e. C -> dom ( A |` B) e. V)
2	1	adantl 388	. . 3 |- ((Fun A /\ B e. C) -> dom ( A |` B) e. V)
3		funimaexg 3575	. . . 4 |- ((Fun A /\ B e. C) -> (A"B) e. V)
4		df-ima 3191	. . . 4 |- (A"B) = ran ( A |` B)
5	3, 4	syl5eqelr 1553	. . 3 |- ((Fun A /\ B e. C) -> ran ( A |` B) e. V)
6	2, 5	jca 288	. 2 |- ((Fun A /\ B e. C) -> (dom ( A |` B) e. V /\ ran ( A |` B) e. V))
7		xpexg 3259	. 2 |- ((dom ( A |` B) e. V /\ ran ( A |` B) e. V) -> (dom ( A |` B) X. ran ( A |` B)) e. V)
8		relres 3387	. . . 4 |- Rel (A |` B)
9		relssdr 3513	. . . 4 |- (Rel (A |` B) -> (A |` B) (_ (dom ( A |` B) X. ran ( A |` B)))
10	8, 9	ax-mp 7	. . 3 |- (A |` B) (_ (dom ( A |` B) X. ran ( A |` B))
11		ssexg 2721	. . 3 |- (((A |` B) (_ (dom ( A |` B) X. ran ( A |` B)) /\ (dom ( A |` B) X. ran ( A |` B)) e. V) -> (A |` B) e. V)
12	10, 11	mpan 695	. 2 |- ((dom ( A |` B) X. ran ( A |` B)) e. V -> (A |` B) e. V)
13	6, 7, 12	3syl 20	1 |- ((Fun A /\ B e. C) -> (A |` B) e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811   (_ wss 2047   X. cxp 3168  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Rel wrel 3175  Fun wfun 3176

This theorem is referenced by:  cofunexg 3580  fvresex 3857  tz7.44-2 3929  tz7.44-3 3930  numthlem 4783  zorn2lem1 4788  imadomg 4806  fac1 6935  facp1t 6936  sumeq2 6985

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-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

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-rex 1650  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-uni 2504  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-ima 3191  df-fun 3192

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