Theorem ofmres 6202

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 6203, allowing it to be used as a function or structure argument. By ofmresval 6151, 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 3206	. . 3 |- A C_ _V
2		ssv 3206	. . 3 |- B C_ _V
3		resmpo 6024	. . 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 6139	. . 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 4943	. 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 2196	. . 3 |- A = A
8		eqid 2196	. . 3 |- B = B
9		vex 2766	. . . 4 |- f e. _V
10		vex 2766	. . . 4 |- g e. _V
11	9	dmex 4933	. . . . . 6 |- dom f e. _V
12	11	inex1 4168	. . . . 5 |- ( dom f i^i dom g ) e. _V
13	12	mptex 5791	. . . 4 |- ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) e. _V
14	5	ovmpt4g 6049	. . . 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 1348	. . 3 |- ( f oF R g ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) )
16	7, 8, 15	mpoeq123i 5989	. 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 2227	1 |- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   i^i cin 3156   C_ wss 3157   |-> cmpt 4095   X. cxp 4662   dom cdm 4664   |` cres 4666   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   oFcof 6137

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139

This theorem is referenced by: (None)