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Theorem ofrn 30369
 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 7293 . . 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 4169 . . 3 (𝐴𝐴) = 𝐴
72, 3, 4, 5, 5, 6off 7398 . 2 (𝜑 → (𝐹f + 𝐺):𝐴𝐶)
87frnd 6493 1 (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3909   × cxp 5525  ran crn 5528  ⟶wf 6323  (class class class)co 7129   ∘f cof 7381 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pr 5302 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7132  df-oprab 7133  df-mpo 7134  df-of 7383 This theorem is referenced by: (None)
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