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Theorem offvalfv 44293
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 6846 . . 3 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
42, 3sylan 580 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ran 𝐹)
5 offvalfv.g . . 3 (𝜑𝐺 Fn 𝐴)
6 fnfvelrn 6846 . . 3 ((𝐺 Fn 𝐴𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
75, 6sylan 580 . 2 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ ran 𝐺)
8 dffn5 6723 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
92, 8sylib 219 . 2 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
10 dffn5 6723 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
115, 10sylib 219 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
121, 4, 7, 9, 11offval2 7420 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cmpt 5143  ran crn 5555   Fn wfn 6349  cfv 6354  (class class class)co 7150  f cof 7401
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403
This theorem is referenced by:  zlmodzxzscm  44307  zlmodzxzadd  44308  mndpsuppss  44321  lincsum  44386
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