Theorem offvalfv 45630

Description: The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)

Hypotheses

Ref	Expression
offvalfv.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offvalfv.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
offvalfv.g	⊢ (𝜑 → 𝐺 Fn 𝐴)

Assertion

Ref	Expression
offvalfv	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑅

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem offvalfv

Step	Hyp	Ref	Expression
1		offvalfv.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		offvalfv.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fnfvelrn 6952	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
4	2, 3	sylan 579	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
5		offvalfv.g	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
6		fnfvelrn 6952	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
7	5, 6	sylan 579	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
8		dffn5 6822	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	2, 8	sylib 217	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
10		dffn5 6822	. . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
11	5, 10	sylib 217	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
12	1, 4, 7, 9, 11	offval2 7544	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109   ↦ cmpt 5161  ran crn 5589   Fn wfn 6425  ‘cfv 6430  (class class class)co 7268   ∘_f cof 7522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524

This theorem is referenced by:  zlmodzxzscm  45645  zlmodzxzadd  45646  mndpsuppss  45659  lincsum  45722