Theorem ofres 6231

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 2230	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2230	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	offval 6224	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
9		inss1 3424	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
10	5, 9	eqsstrri 3257	. . . 4 ⊢ 𝐶 ⊆ 𝐴
11		fnssres 5435	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶)
12	1, 10, 11	sylancl 413	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶)
13		inss2 3425	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
14	5, 13	eqsstrri 3257	. . . 4 ⊢ 𝐶 ⊆ 𝐵
15		fnssres 5435	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶)
16	2, 14, 15	sylancl 413	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶)
17		ssexg 4222	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V)
18	10, 3, 17	sylancr 414	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
19		inidm 3413	. . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶
20		fvres 5650	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
21	20	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
22		fvres 5650	. . . 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 6224	. 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
25	8, 24	eqtr4d 2265	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∩ cin 3196   ⊆ wss 3197   ↦ cmpt 4144   ↾ cres 4720   Fn wfn 5312  ‘cfv 5317  (class class class)co 6000   ∘_𝑓 cof 6214

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-of 6216

This theorem is referenced by: (None)