Theorem offn 7633

Description: The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
offval.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
offval.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
offval.5	⊢ (𝐴 ∩ 𝐵) = 𝑆

Assertion

Ref	Expression
offn	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆)

Proof of Theorem offn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7389	. . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
2		eqid 2739	. . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
3	1, 2	fnmpti 6628	. 2 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆
4		offval.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		offval.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
6		offval.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		offval.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		offval.5	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆
9		eqidd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	4, 5, 6, 7, 8, 9, 10	offval 7629	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
12	11	fneq1d 6578	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆))
13	3, 12	mpbiri 259	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3882   ↦ cmpt 5153   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356   ∘_f cof 7618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620

This theorem is referenced by:  offun  7634  offveq  7646  suppofss1d  8144  suppofss2d  8145  ofsubeq0  12147  ofnegsub  12148  ofsubge0  12149  seqof  14012  ofccat  14922  frlmsslsp  21771  frlmup1  21773  psrbagcon  21900  psdmul  22154  i1faddlem  25678  i1fmullem  25679  dv11cn  25986  coemulc  26238  ofmulrt  26266  plydivlem3  26279  plyrem  26289  jensen  26970  basellem9  27070  1arithidomlem2  33619  selvply1rhmlemb  33703  mplvrpmrhm  33731  esplyind  33759  ply1degltdimlem  33806  broucube  38021  ofun  42722  fsuppind  43040  ofoafg  43799  ofoafo  43801  ofoaid1  43803  ofoaid2  43804  ofoaass  43805  ofoacom  43806  naddcnff  43807  naddcnffo  43809  naddcnfcom  43811  naddcnfid1  43812  naddcnfass  43814  caofcan  44767  ofmul12  44769  ofdivrec  44770  ofdivcan4  44771  ofdivdiv2  44772  cjnpoly  47352