Theorem resfunexg 5927

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 5413	. . . . 5 |- ( Fun A -> Fun ( A |` B ) )
2		funfvex 5707	. . . . . 6 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> ( ( A |` B ) ` x ) e. _V )
3	2	ralrimiva 2623	. . . . 5 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) ( ( A |` B ) ` x ) e. _V )
4		fnasrng 5880	. . . . 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 5402	. . . . 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 5742	. . . 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 2824	. . . . . . . . 9 |- x e. _V
13		opexg 4363	. . . . . . . . 9 |- ( ( x e. _V /\ ( ( A |` B ) ` x ) e. _V ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
14	12, 2, 13	sylancr 418	. . . . . . . 8 |- ( ( Fun ( A |` B ) /\ x e. dom ( A |` B ) ) -> <. x , ( ( A |` B ) ` x ) >. e. _V )
15	14	ralrimiva 2623	. . . . . . 7 |- ( Fun ( A |` B ) -> A. x e. dom ( A |` B ) <. x , ( ( A |` B ) ` x ) >. e. _V )
16		dmmptg 5280	. . . . . . 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 5121	. . . . 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 5131	. . . . 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 2286	. . . 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 2281	. 2 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) = ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) )
23		funmpt 5410	. . 3 |- Fun ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. )
24		dmresexg 5081	. . . 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 5460	. . 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 418	. 2 |- ( ( Fun A /\ B e. C ) -> ( ( x e. dom ( A |` B ) |-> <. x , ( ( A |` B ) ` x ) >. ) " dom ( A |` B ) ) e. _V )
28	22, 27	eqeltrd 2315	1 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   <.cop 3708   |-> cmpt 4187   dom cdm 4769   ran crn 4770   |` cres 4771   "cima 4772   Fun wfun 5366   Fn wfn 5367   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380

This theorem is referenced by:  fnex  5928  ofexg  6297  cofunexg  6328  rdgivallem  6642  frecex  6655  frecsuclem  6667  djudoml  7565  djudomr  7566  fihashf1rn  11205  qnnen  13300