Theorem resfunexg 5403

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 4961	. . . . 5 |- ( Fun A -> Fun ( A |` B ) )
2		funfvex 5212	. . . . . 6 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> ( ( A |` B ) ` x ) e. _V )
3	2	ralrimiva 2434	. . . . 5 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) ( ( A |` B ) ` x ) e. _V )
4		fnasrng 5364	. . . . 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 270	. . 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 270	. . . . 5 |- ( ( Fun A /\ B e. C ) -> Fun ( A |` B ) )
8		funfn 4951	. . . . 5 |- ( Fun ( A |` B ) <-> ( A |` B ) Fn dom ( A |` B ) )
9	7, 8	sylib 120	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) Fn dom ( A |` B ) )
10		dffn5im 5240	. . . 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 4698	. . . . 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 2604	. . . . . . . . 9 |- x e. _V
14		opexg 3983	. . . . . . . . 9 |- ( ( x e. _V /\ ( ( A |` B ) ` x ) e. _V ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
15	13, 2, 14	sylancr 405	. . . . . . . 8 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
16	15	ralrimiva 2434	. . . . . . 7 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) <. x , ( ( A |` B ) ` x ) >. e. _V )
17		dmmptg 4838	. . . . . . 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 4688	. . . . 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 2128	. . . 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 270	. . 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 2123	. 2 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) = ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) )
23		funmpt 4958	. . 3 |- Fun ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. )
24		dmresexg 4652	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
25	24	adantl 271	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
26		funimaexg 5003	. . 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 405	. 2 |- ( ( Fun A /\ B e. C ) -> ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) e. _V )
28	22, 27	eqeltrd 2155	1 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   <.cop 3401   |-> cmpt 3839   dom cdm 4363   ran crn 4364   |` cres 4365   "cima 4366   Fun wfun 4916   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by:  fnex  5404  ofexg  5736  cofunexg  5758  rdgivallem  5991  frecex  6004  frecsuclem3  6013