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Theorem fnund 6669
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 6668 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
51, 2, 3, 4syl21anc 837 1 (𝜑 → (𝐹𝐺) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  cun 3945  cin 3946  c0 4323   Fn wfn 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-fun 6550  df-fn 6551
This theorem is referenced by:  fnunop  6670  brwdom2  9596  sseqfn  34010  bnj927  34400  metakunt19  41675  metakunt25  41681  ofun  41727  tfsconcatfn  42767
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