Theorem offvalfv 7675

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 7052	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
4	2, 3	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
5		offvalfv.g	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
6		fnfvelrn 7052	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
7	5, 6	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
8		dffn5 6919	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	2, 8	sylib 218	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
10		dffn5 6919	. . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
11	5, 10	sylib 218	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
12	1, 4, 7, 9, 11	offval2 7673	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5188  ran crn 5639   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651

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 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653

This theorem is referenced by:  mndpsuppss  18692  zlmodzxzscm  48345  zlmodzxzadd  48346  lincsum  48418