Theorem offn 7646

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 7402	. . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
2		eqid 2729	. . 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 2730	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
10		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
11	4, 5, 6, 7, 8, 9, 10	offval 7642	. . 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 1540   ∈ wcel 2109   ∩ cin 3910   ↦ cmpt 5183   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∘_f cof 7631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  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 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633

This theorem is referenced by:  offun  7647  offveq  7659  suppofss1d  8160  suppofss2d  8161  ofsubeq0  12159  ofnegsub  12160  ofsubge0  12161  seqof  14000  ofccat  14911  frlmsslsp  21681  frlmup1  21683  psrbagcon  21810  psdmul  22029  i1faddlem  25570  i1fmullem  25571  dv11cn  25882  coemulc  26136  ofmulrt  26165  plydivlem3  26179  plyrem  26189  jensen  26875  basellem9  26975  1arithidomlem2  33480  ply1degltdimlem  33591  broucube  37621  ofun  42197  fsuppind  42551  ofoafg  43316  ofoafo  43318  ofoaid1  43320  ofoaid2  43321  ofoaass  43322  ofoacom  43323  naddcnff  43324  naddcnffo  43326  naddcnfcom  43328  naddcnfid1  43329  naddcnfass  43331  caofcan  44285  ofmul12  44287  ofdivrec  44288  ofdivcan4  44289  ofdivdiv2  44290  cjnpoly  46863