Theorem ofres 7652

Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)

Hypotheses

Ref	Expression
ofres.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofres.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
ofres.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofres.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofres.5	⊢ (𝐴 ∩ 𝐵) = 𝐶

Assertion

Ref	Expression
ofres	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)))

Proof of Theorem ofres

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofres.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		ofres.2	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		ofres.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ofres.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		ofres.5	. . 3 ⊢ (𝐴 ∩ 𝐵) = 𝐶
6		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 7642	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9		inss1 4196	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
10	5, 9	eqsstrri 3991	. . . 4 ⊢ 𝐶 ⊆ 𝐴
11		fnssres 6623	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶)
12	1, 10, 11	sylancl 586	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
13		inss2 4197	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
14	5, 13	eqsstrri 3991	. . . 4 ⊢ 𝐶 ⊆ 𝐵
15		fnssres 6623	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶)
16	2, 14, 15	sylancl 586	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
17		ssexg 5273	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
18	10, 3, 17	sylancr 587	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
19		inidm 4186	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
20		fvres 6859	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
21	20	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
22		fvres 6859	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
23	22	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
24	12, 16, 18, 18, 19, 21, 23	offval 7642	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
25	8, 24	eqtr4d 2767	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911   ↦ cmpt 5183   ↾ cres 5633   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633

This theorem is referenced by:  ofoafg  43336