Theorem ofmres 5705

Description: Equivalent expressions for a restriction of the function operation map. Unlike ∘_𝑓 𝑅 which is a proper class, ( ∘_𝑓 𝑅 ∣ ‘(A × B)) can be a set by ofmresex 5706, allowing it to be used as a function or structure argument. By ofmresval 5665, the restricted operation map values are the same as the original values, allowing theorems for ∘_𝑓 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)

Assertion

Ref	Expression
ofmres	⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g))

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 2959	. . 3 ⊢ A ⊆ V
2		ssv 2959	. . 3 ⊢ B ⊆ V
3		resmpt2 5541	. . 3 ⊢ ((A ⊆ V ∧ B ⊆ V) → ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))))
4	1, 2, 3	mp2an 402	. 2 ⊢ ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))))
5		df-of 5654	. . 3 ⊢ ∘𝑓 𝑅 = (f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))))
6	5	reseq1i 4551	. 2 ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B))
7		eqid 2037	. . 3 ⊢ A = A
8		eqid 2037	. . 3 ⊢ B = B
9		vex 2554	. . . 4 ⊢ f ∈ V
10		vex 2554	. . . 4 ⊢ g ∈ V
11	9	dmex 4541	. . . . . 6 ⊢ dom f ∈ V
12	11	inex1 3882	. . . . 5 ⊢ (dom f ∩ dom g) ∈ V
13	12	mptex 5330	. . . 4 ⊢ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))) ∈ V
14	5	ovmpt4g 5565	. . . 4 ⊢ ((f ∈ V ∧ g ∈ V ∧ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))) ∈ V) → (f ∘𝑓 𝑅g) = (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))))
15	9, 10, 13, 14	mp3an 1231	. . 3 ⊢ (f ∘𝑓 𝑅g) = (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))
16	7, 8, 15	mpt2eq123i 5510	. 2 ⊢ (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g)) = (f ∈ A, g ∈ B ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))))
17	4, 6, 16	3eqtr4i 2067	1 ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g))

Syntax hints:   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∩ cin 2910   ⊆ wss 2911   ↦ cmpt 3809   × cxp 4286  dom cdm 4288   ↾ cres 4290  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457   ∘_𝑓 cof 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654

This theorem is referenced by: (None)