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Theorem ofrn 30955
Description: The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
Hypotheses
Ref Expression
ofrn.1 (𝜑𝐹:𝐴𝐵)
ofrn.2 (𝜑𝐺:𝐴𝐵)
ofrn.3 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
ofrn.4 (𝜑𝐴𝑉)
Assertion
Ref Expression
ofrn (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)

Proof of Theorem ofrn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrn.3 . . . 4 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
21fovrnda 7434 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐶)
3 ofrn.1 . . 3 (𝜑𝐹:𝐴𝐵)
4 ofrn.2 . . 3 (𝜑𝐺:𝐴𝐵)
5 ofrn.4 . . 3 (𝜑𝐴𝑉)
6 inidm 4157 . . 3 (𝐴𝐴) = 𝐴
72, 3, 4, 5, 5, 6off 7542 . 2 (𝜑 → (𝐹f + 𝐺):𝐴𝐶)
87frnd 6604 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2109  wss 3891   × cxp 5586  ran crn 5589  wf 6426  (class class class)co 7268  f cof 7522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524
This theorem is referenced by: (None)
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