Theorem off 7077

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 3976	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	2, 3	eqsstr3i 3777	. . . . . 6 ⊢ 𝐶 ⊆ 𝐴
5	4	sseli 3740	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴)
6		ffvelrn 6520	. . . . 5 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆)
7	1, 5, 6	syl2an 495	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
8		off.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
9		inss2 3977	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
10	2, 9	eqsstr3i 3777	. . . . . 6 ⊢ 𝐶 ⊆ 𝐵
11	10	sseli 3740	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵)
12		ffvelrn 6520	. . . . 5 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇)
13	8, 11, 12	syl2an 495	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
14		off.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
15	14	ralrimivva 3109	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
16	15	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17		oveq1 6820	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥𝑅𝑦) = ((𝐹‘𝑧)𝑅𝑦))
18	17	eleq1d 2824	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈))
19		oveq2 6821	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑧)𝑅𝑦) = ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
20	19	eleq1d 2824	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈))
21	18, 20	rspc2va 3462	. . . 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 1476	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
23		eqid 2760	. . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
24	22, 23	fmptd 6548	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)
25		ffn 6206	. . . . 5 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
26	1, 25	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27		ffn 6206	. . . . 5 ⊢ (𝐺:𝐵⟶𝑇 → 𝐺 Fn 𝐵)
28	8, 27	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
29		off.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
30		off.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
31		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
32		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
33	26, 28, 29, 30, 2, 31, 32	offval 7069	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
34	33	feq1d 6191	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈))
35	24, 34	mpbird 247	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050   ∩ cin 3714   ↦ cmpt 4881   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ∘_𝑓 cof 7060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062

This theorem is referenced by:  o1of2  14542  ghmplusg  18449  gsumzaddlem  18521  gsumzadd  18522  lcomf  19104  psrbagaddcl  19572  psraddcl  19585  psrvscacl  19595  psrbagev1  19712  evlslem3  19716  frlmup1  20339  mndvcl  20399  tsmsadd  22151  mbfmulc2lem  23613  mbfaddlem  23626  i1fadd  23661  i1fmul  23662  itg1addlem4  23665  i1fmulclem  23668  i1fmulc  23669  mbfi1flimlem  23688  itg2mulclem  23712  itg2mulc  23713  itg2monolem1  23716  itg2addlem  23724  dvaddbr  23900  dvmulbr  23901  dvaddf  23904  dvmulf  23905  dv11cn  23963  plyaddlem  24170  coeeulem  24179  coeaddlem  24204  plydivlem4  24250  jensenlem2  24913  jensen  24914  basellem7  25012  basellem9  25014  dchrmulcl  25173  ofrn  29750  sibfof  30711  signshf  30974  circlemethhgt  31030  poimirlem23  33745  poimirlem24  33746  poimirlem25  33747  poimirlem29  33751  poimirlem30  33752  poimirlem31  33753  poimirlem32  33754  itg2addnc  33777  ftc1anclem3  33800  ftc1anclem6  33803  ftc1anclem8  33805  lfladdcl  34861  lflvscl  34867  mzpclall  37792  mzpindd  37811  expgrowth  39036  binomcxplemnotnn0  39057  dvdivcncf  40645  ofaddmndmap  42632  amgmwlem  43061