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Theorem offvalfv 47592
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 7089 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
42, 3sylan 578 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
5 offvalfv.g . . 3 (𝜑𝐺 Fn 𝐴)
6 fnfvelrn 7089 . . 3 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
75, 6sylan 578 . 2 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
8 dffn5 6956 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
92, 8sylib 217 . 2 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
10 dffn5 6956 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
115, 10sylib 217 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
121, 4, 7, 9, 11offval2 7705 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5232  ran crn 5679   Fn wfn 6544  cfv 6549  (class class class)co 7419  f cof 7683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685
This theorem is referenced by:  zlmodzxzscm  47607  zlmodzxzadd  47608  mndpsuppss  47621  lincsum  47683
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