Theorem ofres 5725

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 2041	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2041	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 5719	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9		inss1 3157	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
10	5, 9	eqsstr3i 2976	. . . 4 ⊢ 𝐶 ⊆ 𝐴
11		fnssres 5012	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶)
12	1, 10, 11	sylancl 392	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
13		inss2 3158	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
14	5, 13	eqsstr3i 2976	. . . 4 ⊢ 𝐶 ⊆ 𝐵
15		fnssres 5012	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶)
16	2, 14, 15	sylancl 392	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
17		ssexg 3896	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
18	10, 3, 17	sylancr 393	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
19		inidm 3146	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
20		fvres 5198	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
21	20	adantl 262	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
22		fvres 5198	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
23	22	adantl 262	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
24	12, 16, 18, 18, 19, 21, 23	offval 5719	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
25	8, 24	eqtr4d 2075	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∩ cin 2916   ⊆ wss 2917   ↦ cmpt 3818   ↾ cres 4347   Fn wfn 4897  ‘cfv 4902  (class class class)co 5512   ∘_𝑓 cof 5710

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712

This theorem is referenced by: (None)