Theorem resfunexg 5325

Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)

Assertion

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

Proof of Theorem resfunexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funres 4884	. . . . 5 |- ( Fun A -> Fun (A |` B))
2		funfvex 5135	. . . . . 6 |- (( Fun (A |` B) /\ x e. dom (A |` B)) -> ((A |` B) ` x) e. _V)
3	2	ralrimiva 2386	. . . . 5 |- ( Fun (A |` B) -> A.x e. dom (A |` B)((A |` B) ` x) e. _V)
4		fnasrng 5286	. . . . 5 |- (A.x e. dom (A |` B)((A |` B) ` x) e. _V -> (x e. dom (A |` B) |-> ((A |` B) ` x)) = ran (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.))
5	1, 3, 4	3syl 17	. . . 4 |- ( Fun A -> (x e. dom (A |` B) |-> ((A |` B) ` x)) = ran (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.))
6	5	adantr 261	. . 3 |- (( Fun A /\ B e. C) -> (x e. dom (A |` B) |-> ((A |` B) ` x)) = ran (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.))
7	1	adantr 261	. . . . 5 |- (( Fun A /\ B e. C) -> Fun (A |` B))
8		funfn 4874	. . . . 5 |- ( Fun (A |` B) <-> (A |` B) Fn dom (A |` B))
9	7, 8	sylib 127	. . . 4 |- (( Fun A /\ B e. C) -> (A |` B) Fn dom (A |` B))
10		dffn5im 5162	. . . 4 |- ((A |` B) Fn dom (A |` B) -> (A |` B) = (x e. dom (A |` B) |-> ((A |` B) ` x)))
11	9, 10	syl 14	. . 3 |- (( Fun A /\ B e. C) -> (A |` B) = (x e. dom (A |` B) |-> ((A |` B) ` x)))
12		imadmrn 4621	. . . . 5 |- ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.)) = ran (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.)
13		vex 2554	. . . . . . . . 9 |- x e. _V
14		opexgOLD 3956	. . . . . . . . 9 |- ((x e. _V /\ ((A |` B) ` x) e. _V) -> <.x , ((A |` B) ` x) >. e. _V)
15	13, 2, 14	sylancr 393	. . . . . . . 8 |- (( Fun (A |` B) /\ x e. dom (A |` B)) -> <.x , ((A |` B) ` x) >. e. _V)
16	15	ralrimiva 2386	. . . . . . 7 |- ( Fun (A |` B) -> A.x e. dom (A |` B) <.x , ((A |` B) ` x) >. e. _V)
17		dmmptg 4761	. . . . . . 7 |- (A.x e. dom (A |` B) <.x , ((A |` B) ` x) >. e. _V -> dom (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) = dom (A |` B))
18	1, 16, 17	3syl 17	. . . . . 6 |- ( Fun A -> dom (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) = dom (A |` B))
19	18	imaeq2d 4611	. . . . 5 |- ( Fun A -> ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.)) = ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (A |` B)))
20	12, 19	syl5reqr 2084	. . . 4 |- ( Fun A -> ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (A |` B)) = ran (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.))
21	20	adantr 261	. . 3 |- (( Fun A /\ B e. C) -> ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (A |` B)) = ran (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.))
22	6, 11, 21	3eqtr4d 2079	. 2 |- (( Fun A /\ B e. C) -> (A |` B) = ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (A |` B)))
23		funmpt 4881	. . 3 |- Fun (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.)
24		dmresexg 4577	. . . 4 |- (B e. C -> dom (A |` B) e. _V)
25	24	adantl 262	. . 3 |- (( Fun A /\ B e. C) -> dom (A |` B) e. _V)
26		funimaexg 4926	. . 3 |- (( Fun (x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) /\ dom (A |` B) e. _V) -> ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (A |` B)) e. _V)
27	23, 25, 26	sylancr 393	. 2 |- (( Fun A /\ B e. C) -> ((x e. dom (A |` B) |-> <.x , ((A |` B) ` x) >.) " dom (A |` B)) e. _V)
28	22, 27	eqeltrd 2111	1 |- (( Fun A /\ B e. C) -> (A |` B) e. _V)

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390  A.wral 2300   _Vcvv 2551   <.cop 3370   |-> cmpt 3809   dom cdm 4288   ran crn 4289   |` cres 4290   "cima 4291   Fun wfun 4839   Fn wfn 4840   ` cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  fnex  5326  ofexg  5658  cofunexg  5680  rdgivallem  5908  frecsuclem3  5929