Theorem ofmres 6034

Description: Equivalent expressions for a restriction of the function operation map. Unlike oF R which is a proper class, ( oF R |` ( A X. B ) ) can be a set by ofmresex 6035, allowing it to be used as a function or structure argument. By ofmresval 5993, the restricted operation map values are the same as the original values, allowing theorems for oF R to be reused. (Contributed by NM, 20-Oct-2014.)

Assertion

Ref	Expression
ofmres	|- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )

Distinct variable groups:   f, g, A   B, f, g   R, f, g

Proof of Theorem ofmres

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3119	. . 3 |- A C_ _V
2		ssv 3119	. . 3 |- B C_ _V
3		resmpo 5869	. . 3 |- ( ( A C_ _V /\ B C_ _V ) -> ( ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) |` ( A X. B ) ) = ( f e. A , g e. B |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) )
4	1, 2, 3	mp2an 422	. 2 |- ( ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) |` ( A X. B ) ) = ( f e. A , g e. B |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
5		df-of 5982	. . 3 |- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
6	5	reseq1i 4815	. 2 |- ( oF R |` ( A X. B ) ) = ( ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) |` ( A X. B ) )
7		eqid 2139	. . 3 |- A = A
8		eqid 2139	. . 3 |- B = B
9		vex 2689	. . . 4 |- f e. _V
10		vex 2689	. . . 4 |- g e. _V
11	9	dmex 4805	. . . . . 6 |- dom f e. _V
12	11	inex1 4062	. . . . 5 |- ( dom f i^i dom g ) e. _V
13	12	mptex 5646	. . . 4 |- ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) e. _V
14	5	ovmpt4g 5893	. . . 4 |- ( ( f e. _V /\ g e. _V /\ ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) e. _V ) -> ( f oF R g ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
15	9, 10, 13, 14	mp3an 1315	. . 3 |- ( f oF R g ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) )
16	7, 8, 15	mpoeq123i 5834	. 2 |- ( f e. A , g e. B |-> ( f oF R g ) ) = ( f e. A , g e. B |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
17	4, 6, 16	3eqtr4i 2170	1 |- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2686   i^i cin 3070   C_ wss 3071   |-> cmpt 3989   X. cxp 4537   dom cdm 4539   |` cres 4541   ` cfv 5123  (class class class)co 5774   e. cmpo 5776   oFcof 5980

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-in1 603  ax-in2 604  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-13 1491  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  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  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-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  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  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982

This theorem is referenced by: (None)