Theorem ofmres 5686

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 5687, allowing it to be used as a function or structure argument. By ofmresval 5646, 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 2942	. . 3 ⊢ A ⊆ V
2		ssv 2942	. . 3 ⊢ B ⊆ V
3		resmpt2 5522	. . 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 404	. 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 5635	. . 3 ⊢ ∘𝑓 𝑅 = (f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))))
6	5	reseq1i 4535	. 2 ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = ((f ∈ V, g ∈ V ↦ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))) ↾ (A × B))
7		eqid 2022	. . 3 ⊢ A = A
8		eqid 2022	. . 3 ⊢ B = B
9		vex 2538	. . . 4 ⊢ f ∈ V
10		vex 2538	. . . 4 ⊢ g ∈ V
11	9	dmex 4525	. . . . . 6 ⊢ dom f ∈ V
12	11	inex1 3865	. . . . 5 ⊢ (dom f ∩ dom g) ∈ V
13	12	mptex 5312	. . . 4 ⊢ (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x))) ∈ V
14	5	ovmpt4g 5546	. . . 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 1217	. . 3 ⊢ (f ∘𝑓 𝑅g) = (x ∈ (dom f ∩ dom g) ↦ ((f‘x)𝑅(g‘x)))
16	7, 8, 15	mpt2eq123i 5491	. 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 2052	1 ⊢ ( ∘𝑓 𝑅 ↾ (A × B)) = (f ∈ A, g ∈ B ↦ (f ∘𝑓 𝑅g))

Syntax hints:   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ∩ cin 2893   ⊆ wss 2894   ↦ cmpt 3792   × cxp 4270  dom cdm 4272   ↾ cres 4274  ‘cfv 4829  (class class class)co 5436   ↦ cmpt2 5438   ∘_𝑓 cof 5633

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-of 5635

This theorem is referenced by: (None)