Theorem ofmres 6115

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 6116, allowing it to be used as a function or structure argument. By ofmresval 6072, 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 3169	. . 3 |- A C_ _V
2		ssv 3169	. . 3 |- B C_ _V
3		resmpo 5951	. . 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 424	. 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 6061	. . 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 4887	. 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 2170	. . 3 |- A = A
8		eqid 2170	. . 3 |- B = B
9		vex 2733	. . . 4 |- f e. _V
10		vex 2733	. . . 4 |- g e. _V
11	9	dmex 4877	. . . . . 6 |- dom f e. _V
12	11	inex1 4123	. . . . 5 |- ( dom f i^i dom g ) e. _V
13	12	mptex 5722	. . . 4 |- ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) e. _V
14	5	ovmpt4g 5975	. . . 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 1332	. . 3 |- ( f oF R g ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) )
16	7, 8, 15	mpoeq123i 5916	. 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 2201	1 |- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   i^i cin 3120   C_ wss 3121   |-> cmpt 4050   X. cxp 4609   dom cdm 4611   |` cres 4613   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   oFcof 6059

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061

This theorem is referenced by: (None)