Theorem ofmres 6139

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 6140, allowing it to be used as a function or structure argument. By ofmresval 6096, 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 3179	. . 3 |- A C_ _V
2		ssv 3179	. . 3 |- B C_ _V
3		resmpo 5975	. . 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 426	. 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 6085	. . 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 4905	. 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 2177	. . 3 |- A = A
8		eqid 2177	. . 3 |- B = B
9		vex 2742	. . . 4 |- f e. _V
10		vex 2742	. . . 4 |- g e. _V
11	9	dmex 4895	. . . . . 6 |- dom f e. _V
12	11	inex1 4139	. . . . 5 |- ( dom f i^i dom g ) e. _V
13	12	mptex 5744	. . . 4 |- ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) e. _V
14	5	ovmpt4g 5999	. . . 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 1337	. . 3 |- ( f oF R g ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) )
16	7, 8, 15	mpoeq123i 5940	. 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 2208	1 |- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2739   i^i cin 3130   C_ wss 3131   |-> cmpt 4066   X. cxp 4626   dom cdm 4628   |` cres 4630   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   oFcof 6083

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-in1 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085

This theorem is referenced by: (None)