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Theorem fnund 6683
Description: The union of two functions with disjoint domains, a deduction version. (Contributed by metakunt, 28-May-2024.)
Hypotheses
Ref Expression
fnund.1 (𝜑𝐹 Fn 𝐴)
fnund.2 (𝜑𝐺 Fn 𝐵)
fnund.3 (𝜑 → (𝐴𝐵) = ∅)
Assertion
Ref Expression
fnund (𝜑 → (𝐹𝐺) Fn (𝐴𝐵))

Proof of Theorem fnund
StepHypRef Expression
1 fnund.1 . 2 (𝜑𝐹 Fn 𝐴)
2 fnund.2 . 2 (𝜑𝐺 Fn 𝐵)
3 fnund.3 . 2 (𝜑 → (𝐴𝐵) = ∅)
4 fnun 6682 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
51, 2, 3, 4syl21anc 838 1 (𝜑 → (𝐹𝐺) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cun 3949  cin 3950  c0 4333   Fn wfn 6556
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-12 2177  ax-ext 2708  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-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  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-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563  df-fn 6564
This theorem is referenced by:  fnunop  6684  brwdom2  9613  sseqfn  34392  bnj927  34783  metakunt19  42224  metakunt25  42230  ofun  42277  tfsconcatfn  43351
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