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Theorem ofrn 32336
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 7572 . . 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 4211 . . 3 (𝐴𝐴) = 𝐴
72, 3, 4, 5, 5, 6off 7682 . 2 (𝜑 → (𝐹f + 𝐺):𝐴𝐶)
87frnd 6716 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3941   × cxp 5665  ran crn 5668  wf 6530  (class class class)co 7402  f cof 7662
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664
This theorem is referenced by: (None)
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