Theorem offn 7644

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 7400	. . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
2		eqid 2736	. . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
3	1, 2	fnmpti 6641	. 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 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	4, 5, 6, 7, 8, 9, 10	offval 7640	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
12	11	fneq1d 6591	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆))
13	3, 12	mpbiri 258	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3888   ↦ cmpt 5166   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631

This theorem is referenced by:  offun  7645  offveq  7657  suppofss1d  8154  suppofss2d  8155  ofsubeq0  12156  ofnegsub  12157  ofsubge0  12158  seqof  14021  ofccat  14931  frlmsslsp  21776  frlmup1  21778  psrbagcon  21905  psdmul  22132  i1faddlem  25660  i1fmullem  25661  dv11cn  25968  coemulc  26220  ofmulrt  26248  plydivlem3  26261  plyrem  26271  jensen  26952  basellem9  27052  1arithidomlem2  33596  mplvrpmrhm  33691  esplyind  33719  ply1degltdimlem  33766  broucube  37975  ofun  42677  fsuppind  43023  ofoafg  43782  ofoafo  43784  ofoaid1  43786  ofoaid2  43787  ofoaass  43788  ofoacom  43789  naddcnff  43790  naddcnffo  43792  naddcnfcom  43794  naddcnfid1  43795  naddcnfass  43797  caofcan  44750  ofmul12  44752  ofdivrec  44753  ofdivcan4  44754  ofdivdiv2  44755  cjnpoly  47337