Theorem ofres 6173

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 2206	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2206	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 6166	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9		inss1 3393	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
10	5, 9	eqsstrri 3226	. . . 4 ⊢ 𝐶 ⊆ 𝐴
11		fnssres 5389	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶)
12	1, 10, 11	sylancl 413	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
13		inss2 3394	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
14	5, 13	eqsstrri 3226	. . . 4 ⊢ 𝐶 ⊆ 𝐵
15		fnssres 5389	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶)
16	2, 14, 15	sylancl 413	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
17		ssexg 4183	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
18	10, 3, 17	sylancr 414	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
19		inidm 3382	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
20		fvres 5600	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
21	20	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
22		fvres 5600	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
23	22	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥))
24	12, 16, 18, 18, 19, 21, 23	offval 6166	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
25	8, 24	eqtr4d 2241	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176  Vcvv 2772   ∩ cin 3165   ⊆ wss 3166   ↦ cmpt 4105   ↾ cres 4677   Fn wfn 5266  ‘cfv 5271  (class class class)co 5944   ∘_𝑓 cof 6156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-of 6158

This theorem is referenced by: (None)