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Theorem offvalfv 42446
 Description: The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
Hypotheses
Ref Expression
offvalfv.a (𝜑𝐴𝑉)
offvalfv.f (𝜑𝐹 Fn 𝐴)
offvalfv.g (𝜑𝐺 Fn 𝐴)
Assertion
Ref Expression
offvalfv (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑅
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem offvalfv
StepHypRef Expression
1 offvalfv.a . 2 (𝜑𝐴𝑉)
2 offvalfv.f . . 3 (𝜑𝐹 Fn 𝐴)
3 fnfvelrn 6396 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
42, 3sylan 487 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
5 offvalfv.g . . 3 (𝜑𝐺 Fn 𝐴)
6 fnfvelrn 6396 . . 3 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
75, 6sylan 487 . 2 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
8 dffn5 6280 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
92, 8sylib 208 . 2 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
10 dffn5 6280 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
115, 10sylib 208 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
121, 4, 7, 9, 11offval2 6956 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ↦ cmpt 4762  ran crn 5144   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939 This theorem is referenced by:  zlmodzxzscm  42460  zlmodzxzadd  42461  mndpsuppss  42477  lincsum  42543
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