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Theorem fnund 6651
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 6650 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺) Fn (𝐴𝐵))
51, 2, 3, 4syl21anc 836 1 (𝜑 → (𝐹𝐺) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cun 3942  cin 3943  c0 4318   Fn wfn 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-fun 6534  df-fn 6535
This theorem is referenced by:  fnunop  6652  brwdom2  9550  sseqfn  33220  bnj927  33611  metakunt19  40808  metakunt25  40814  ofun  40870  tfsconcatfn  41859
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