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Theorem fnund 6443
 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 6442 . 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 1538   ∪ cun 3881   ∩ cin 3882  ∅c0 4246   Fn wfn 6327 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-id 5429  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-fun 6334  df-fn 6335 This theorem is referenced by:  fnunsn  6444  brwdom2  9039  sseqfn  31824  bnj927  32216  metakunt19  39519  metakunt25  39525  ofun  39569
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