Theorem ofmres 6281

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 6282, allowing it to be used as a function or structure argument. By ofmresval 6230, 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 3246	. . 3 |- A C_ _V
2		ssv 3246	. . 3 |- B C_ _V
3		resmpo 6102	. . 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 6218	. . 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 5001	. 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 2229	. . 3 |- A = A
8		eqid 2229	. . 3 |- B = B
9		vex 2802	. . . 4 |- f e. _V
10		vex 2802	. . . 4 |- g e. _V
11	9	dmex 4991	. . . . . 6 |- dom f e. _V
12	11	inex1 4218	. . . . 5 |- ( dom f i^i dom g ) e. _V
13	12	mptex 5865	. . . 4 |- ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) e. _V
14	5	ovmpt4g 6127	. . . 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 1371	. . 3 |- ( f oF R g ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) )
16	7, 8, 15	mpoeq123i 6067	. 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 2260	1 |- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   C_ wss 3197   |-> cmpt 4145   X. cxp 4717   dom cdm 4719   |` cres 4721   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   oFcof 6216

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-of 6218

This theorem is referenced by: (None)