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Theorem ofrn 32655
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)
21fovcdmda 7603 . . 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 4234 . . 3 (𝐴𝐴) = 𝐴
72, 3, 4, 5, 5, 6off 7714 . 2 (𝜑 → (𝐹f + 𝐺):𝐴𝐶)
87frnd 6744 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3962   × cxp 5686  ran crn 5689  wf 6558  (class class class)co 7430  f cof 7694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696
This theorem is referenced by: (None)
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