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Theorem offvalfv 42908
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 6580 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
42, 3sylan 576 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
5 offvalfv.g . . 3 (𝜑𝐺 Fn 𝐴)
6 fnfvelrn 6580 . . 3 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
75, 6sylan 576 . 2 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
8 dffn5 6464 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
92, 8sylib 210 . 2 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
10 dffn5 6464 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
115, 10sylib 210 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
121, 4, 7, 9, 11offval2 7146 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cmpt 4920  ran crn 5311   Fn wfn 6094  cfv 6099  (class class class)co 6876  𝑓 cof 7127
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-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
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 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-of 7129
This theorem is referenced by:  zlmodzxzscm  42922  zlmodzxzadd  42923  mndpsuppss  42939  lincsum  43005
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