Theorem offn 7645

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 7401	. . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
2		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
3	1, 2	fnmpti 6643	. 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 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	4, 5, 6, 7, 8, 9, 10	offval 7641	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
12	11	fneq1d 6593	. 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 3902   ↦ cmpt 5181   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   ∘_f cof 7630

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632

This theorem is referenced by:  offun  7646  offveq  7658  suppofss1d  8156  suppofss2d  8157  ofsubeq0  12154  ofnegsub  12155  ofsubge0  12156  seqof  13994  ofccat  14904  frlmsslsp  21763  frlmup1  21765  psrbagcon  21893  psdmul  22121  i1faddlem  25662  i1fmullem  25663  dv11cn  25974  coemulc  26228  ofmulrt  26257  plydivlem3  26271  plyrem  26281  jensen  26967  basellem9  27067  1arithidomlem2  33629  mplvrpmrhm  33724  esplyind  33752  ply1degltdimlem  33800  broucube  37905  ofun  42608  fsuppind  42948  ofoafg  43711  ofoafo  43713  ofoaid1  43715  ofoaid2  43716  ofoaass  43717  ofoacom  43718  naddcnff  43719  naddcnffo  43721  naddcnfcom  43723  naddcnfid1  43724  naddcnfass  43726  caofcan  44679  ofmul12  44681  ofdivrec  44682  ofdivcan4  44683  ofdivdiv2  44684  cjnpoly  47249