Theorem resfunexg 5641

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 5164	. . . . 5 |- ( Fun A -> Fun ( A |` B ) )
2		funfvex 5438	. . . . . 6 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> ( ( A |` B ) ` x ) e. _V )
3	2	ralrimiva 2505	. . . . 5 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) ( ( A |` B ) ` x ) e. _V )
4		fnasrng 5600	. . . . 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 274	. . 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 274	. . . . 5 |- ( ( Fun A /\ B e. C ) -> Fun ( A |` B ) )
8		funfn 5153	. . . . 5 |- ( Fun ( A |` B ) <-> ( A |` B ) Fn dom ( A |` B ) )
9	7, 8	sylib 121	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) Fn dom ( A |` B ) )
10		dffn5im 5467	. . . 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 4891	. . . . 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 2689	. . . . . . . . 9 |- x e. _V
14		opexg 4150	. . . . . . . . 9 |- ( ( x e. _V /\ ( ( A |` B ) ` x ) e. _V ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
15	13, 2, 14	sylancr 410	. . . . . . . 8 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
16	15	ralrimiva 2505	. . . . . . 7 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) <. x , ( ( A |` B ) ` x ) >. e. _V )
17		dmmptg 5036	. . . . . . 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 4881	. . . . 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 2187	. . . 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 274	. . 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 2182	. 2 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) = ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) )
23		funmpt 5161	. . 3 |- Fun ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. )
24		dmresexg 4842	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
25	24	adantl 275	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
26		funimaexg 5207	. . 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 410	. 2 |- ( ( Fun A /\ B e. C ) -> ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) e. _V )
28	22, 27	eqeltrd 2216	1 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2416   _Vcvv 2686   <.cop 3530   |-> cmpt 3989   dom cdm 4539   ran crn 4540   |` cres 4541   "cima 4542   Fun wfun 5117   Fn wfn 5118   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131

This theorem is referenced by:  fnex  5642  ofexg  5986  cofunexg  6009  rdgivallem  6278  frecex  6291  frecsuclem  6303  djudoml  7075  djudomr  7076  fihashf1rn  10535  qnnen  11944