Theorem resfunexg 2717

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

Assertion

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

Proof of Theorem resfunexg

Step	Hyp	Ref	Expression
1		dmresexg 2586	. . . . 5 |- (B e. C -> dom (A |` B) e. V)
2	1	adantr 306	. . . 4 |- ((B e. C /\ Fun A) -> dom (A |` B) e. V)
3		funimaexg 2715	. . . . . 6 |- (B e. C -> (Fun A -> (A"B) e. V))
4		df-ima 2431	. . . . . . 7 |- (A"B) = ran (A |` B)
5	4	eleq1i 1152	. . . . . 6 |- ((A"B) e. V <-> ran (A |` B) e. V)
6	3, 5	syl6ib 185	. . . . 5 |- (B e. C -> (Fun A -> ran (A |` B) e. V))
7	6	imp 277	. . . 4 |- ((B e. C /\ Fun A) -> ran (A |` B) e. V)
8	2, 7	jca 236	. . 3 |- ((B e. C /\ Fun A) -> (dom (A |` B) e. V /\ ran (A |` B) e. V))
9		xpexg 2489	. . 3 |- ((dom (A |` B) e. V /\ ran (A |` B) e. V) -> (dom (A |` B) X. ran (A |` B)) e. V)
10		relres 2591	. . . . 5 |- Rel (A |` B)
11		relssdr 2668	. . . . 5 |- (Rel (A |` B) -> (A |` B) (_ (dom (A |` B) X. ran (A |` B)))
12	10, 11	ax-mp 6	. . . 4 |- (A |` B) (_ (dom (A |` B) X. ran (A |` B))
13		ssexg 1702	. . . 4 |- ((dom (A |` B) X. ran (A |` B)) e. V -> ((A |` B) (_ (dom (A |` B) X. ran (A |` B)) -> (A |` B) e. V))
14	12, 13	mpi 44	. . 3 |- ((dom (A |` B) X. ran (A |` B)) e. V -> (A |` B) e. V)
15	8, 9, 14	3syl 21	. 2 |- ((B e. C /\ Fun A) -> (A |` B) e. V)
16	15	exp 291	1 |- (B e. C -> (Fun A -> (A |` B) e. V))

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  Vcvv 1348   (_ wss 1487   X. cxp 2408  dom cdm 2410  ran crn 2411   |` cres 2412  "cima 2413  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  fvresex 2909  tz7.44-2 2967  tz7.44-3 2968  enrefg 3294  numthlem 3598  zornlem1 3603  imadomg 3616  fac0 4871  fac1 4872  facp1t 4873

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-un 1076  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-rex 1206  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-uni 1920  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-ima 2431  df-fun 2432

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