Theorem ofres 6913

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 2623	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2623	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 6904	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9		inss1 3833	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
10	5, 9	eqsstr3i 3636	. . . 4 ⊢ 𝐶 ⊆ 𝐴
11		fnssres 6004	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶)
12	1, 10, 11	sylancl 694	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
13		inss2 3834	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
14	5, 13	eqsstr3i 3636	. . . 4 ⊢ 𝐶 ⊆ 𝐵
15		fnssres 6004	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶)
16	2, 14, 15	sylancl 694	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
17		ssexg 4804	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
18	10, 3, 17	sylancr 695	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
19		inidm 3822	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
20		fvres 6207	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
21	20	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
22		fvres 6207	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
23	22	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
24	12, 16, 18, 18, 19, 21, 23	offval 6904	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
25	8, 24	eqtr4d 2659	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574   ↦ cmpt 4729   ↾ cres 5116   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by: (None)