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Theorem offvalfv 41382
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 6313 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
42, 3sylan 488 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
5 offvalfv.g . . 3 (𝜑𝐺 Fn 𝐴)
6 fnfvelrn 6313 . . 3 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
75, 6sylan 488 . 2 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
8 dffn5 6199 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
92, 8sylib 208 . 2 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
10 dffn5 6199 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
115, 10sylib 208 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
121, 4, 7, 9, 11offval2 6868 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  cmpt 4678  ran crn 5080   Fn wfn 5845  cfv 5850  (class class class)co 6605  𝑓 cof 6849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851
This theorem is referenced by:  zlmodzxzscm  41396  zlmodzxzadd  41397  mndpsuppss  41414  lincsum  41480
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