Theorem off 6229

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 3424	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	2, 3	eqsstrri 3257	. . . . . 6 ⊢ 𝐶 ⊆ 𝐴
5	4	sseli 3220	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴)
6		ffvelcdm 5767	. . . . 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 3425	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
10	2, 9	eqsstrri 3257	. . . . . 6 ⊢ 𝐶 ⊆ 𝐵
11	10	sseli 3220	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵)
12		ffvelcdm 5767	. . . . 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 2612	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
16	15	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17		oveq1 6007	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥𝑅𝑦) = ((𝐹‘𝑧)𝑅𝑦))
18	17	eleq1d 2298	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈))
19		oveq2 6008	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑧)𝑅𝑦) = ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
20	19	eleq1d 2298	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈))
21	18, 20	rspc2va 2921	. . . 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 1270	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
23		eqid 2229	. . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
24	22, 23	fmptd 5788	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)
25		ffn 5472	. . . . 5 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
26	1, 25	syl 14	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27		ffn 5472	. . . . 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 2230	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
32		eqidd 2230	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
33	26, 28, 29, 30, 2, 31, 32	offval 6224	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
34	33	feq1d 5459	. 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 1395   ∈ wcel 2200  ∀wral 2508   ∩ cin 3196   ↦ cmpt 4144   Fn wfn 5312  ⟶wf 5313  ‘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:  offeq  6230  ofnegsub  9105  lcomf  14285  psraddcl  14638  mplsubgfilemcl  14657  dvaddxxbr  15369  dvmulxxbr  15370  dvaddxx  15371  dvmulxx  15372  dviaddf  15373  dvimulf  15374  plyaddlem  15417