Theorem offn 7142

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 6910	. . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
2		eqid 2799	. . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
3	1, 2	fnmpti 6233	. 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 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	4, 5, 6, 7, 8, 9, 10	offval 7138	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
12	11	fneq1d 6192	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆))
13	3, 12	mpbiri 250	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝑆)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ∩ cin 3768   ↦ cmpt 4922   Fn wfn 6096  ‘cfv 6101  (class class class)co 6878   ∘_𝑓 cof 7129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-of 7131

This theorem is referenced by:  offveq  7152  suppofss1d  7570  suppofss2d  7571  ofsubeq0  11309  ofnegsub  11310  ofsubge0  11311  seqof  13112  ofccat  14051  lcomfsupp  19221  psrbagcon  19694  psrbagev1  19832  frlmsslsp  20460  frlmup1  20462  i1faddlem  23801  i1fmullem  23802  dv11cn  24105  coemulc  24352  ofmulrt  24378  plydivlem3  24391  plyrem  24401  jensen  25067  basellem9  25167  broucube  33932  caofcan  39304  ofmul12  39306  ofdivrec  39307  ofdivcan4  39308  ofdivdiv2  39309  mndpsuppss  42951  mndpfsupp  42956