Theorem off 6183

Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
off.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
off.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
off.3	⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
off.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
off.5	⊢ (𝜑 → 𝐵 ∈ 𝑊)
off.6	⊢ (𝐴 ∩ 𝐵) = 𝐶

Assertion

Ref	Expression
off	⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)

Distinct variable groups:   𝑦,𝐺   𝑥,𝑦,𝜑   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem off

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		off.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
2		off.6	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) = 𝐶
3		inss1 3397	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	2, 3	eqsstrri 3230	. . . . . 6 ⊢ 𝐶 ⊆ 𝐴
5	4	sseli 3193	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴)
6		ffvelcdm 5725	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆)
7	1, 5, 6	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
8		off.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
9		inss2 3398	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
10	2, 9	eqsstrri 3230	. . . . . 6 ⊢ 𝐶 ⊆ 𝐵
11	10	sseli 3193	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵)
12		ffvelcdm 5725	. . . . 5 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇)
13	8, 11, 12	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
14		off.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
15	14	ralrimivva 2589	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
16	15	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17		oveq1 5963	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥𝑅𝑦) = ((𝐹‘𝑧)𝑅𝑦))
18	17	eleq1d 2275	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈))
19		oveq2 5964	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑧)𝑅𝑦) = ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
20	19	eleq1d 2275	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈))
21	18, 20	rspc2va 2895	. . . 4 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
22	7, 13, 16, 21	syl21anc 1249	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
23		eqid 2206	. . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
24	22, 23	fmptd 5746	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)
25		ffn 5434	. . . . 5 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
26	1, 25	syl 14	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27		ffn 5434	. . . . 5 ⊢ (𝐺:𝐵⟶𝑇 → 𝐺 Fn 𝐵)
28	8, 27	syl 14	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
29		off.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
30		off.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
31		eqidd 2207	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
32		eqidd 2207	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
33	26, 28, 29, 30, 2, 31, 32	offval 6178	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
34	33	feq1d 5421	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈))
35	24, 34	mpbird 167	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485   ∩ cin 3169   ↦ cmpt 4112   Fn wfn 5274  ⟶wf 5275  ‘cfv 5279  (class class class)co 5956   ∘_𝑓 cof 6168

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-setind 4592

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-of 6170

This theorem is referenced by:  offeq  6184  ofnegsub  9050  lcomf  14159  psraddcl  14512  mplsubgfilemcl  14531  dvaddxxbr  15243  dvmulxxbr  15244  dvaddxx  15245  dvmulxx  15246  dviaddf  15247  dvimulf  15248  plyaddlem  15291