Theorem resfunexg 5737

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 5257	. . . . 5 |- ( Fun A -> Fun ( A |` B ) )
2		funfvex 5532	. . . . . 6 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> ( ( A |` B ) ` x ) e. _V )
3	2	ralrimiva 2550	. . . . 5 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) ( ( A |` B ) ` x ) e. _V )
4		fnasrng 5696	. . . . 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 276	. . 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 276	. . . . 5 |- ( ( Fun A /\ B e. C ) -> Fun ( A |` B ) )
8		funfn 5246	. . . . 5 |- ( Fun ( A |` B ) <-> ( A |` B ) Fn dom ( A |` B ) )
9	7, 8	sylib 122	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) Fn dom ( A |` B ) )
10		dffn5im 5561	. . . 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		vex 2740	. . . . . . . . 9 |- x e. _V
13		opexg 4228	. . . . . . . . 9 |- ( ( x e. _V /\ ( ( A |` B ) ` x ) e. _V ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
14	12, 2, 13	sylancr 414	. . . . . . . 8 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
15	14	ralrimiva 2550	. . . . . . 7 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) <. x , ( ( A |` B ) ` x ) >. e. _V )
16		dmmptg 5126	. . . . . . 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 ) )
17	1, 15, 16	3syl 17	. . . . . 6 |- ( Fun A -> dom ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) = dom ( A |` B ) )
18	17	imaeq2d 4970	. . . . 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 ) ) )
19		imadmrn 4980	. . . . 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 ) >. )
20	18, 19	eqtr3di 2225	. . . 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 276	. . 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 2220	. 2 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) = ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) )
23		funmpt 5254	. . 3 |- Fun ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. )
24		dmresexg 4930	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
25	24	adantl 277	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
26		funimaexg 5300	. . 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 414	. 2 |- ( ( Fun A /\ B e. C ) -> ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) e. _V )
28	22, 27	eqeltrd 2254	1 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2737   <.cop 3595   |-> cmpt 4064   dom cdm 4626   ran crn 4627   |` cres 4628   "cima 4629   Fun wfun 5210   Fn wfn 5211   ` cfv 5216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224

This theorem is referenced by:  fnex  5738  ofexg  6086  cofunexg  6109  rdgivallem  6381  frecex  6394  frecsuclem  6406  djudoml  7217  djudomr  7218  fihashf1rn  10767  qnnen  12431