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Theorem ofrn 32649
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 7604 . . 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 4227 . . 3 (𝐴𝐴) = 𝐴
72, 3, 4, 5, 5, 6off 7715 . 2 (𝜑 → (𝐹f + 𝐺):𝐴𝐶)
87frnd 6744 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951   × cxp 5683  ran crn 5686  wf 6557  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by: (None)
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