Theorem fnund 6596

Description: The union of two functions with disjoint domains, a deduction version. (Contributed by metakunt, 28-May-2024.)

Hypotheses

Ref	Expression
fnund.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnund.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
fnund.3	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
fnund	⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))

Proof of Theorem fnund

Step	Hyp	Ref	Expression
1		fnund.1	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fnund.2	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		fnund.3	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
4		fnun 6595	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
5	1, 2, 3, 4	syl21anc 837	1 ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∪ cun 3900   ∩ cin 3901  ∅c0 4283   Fn wfn 6476

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-fun 6483  df-fn 6484

This theorem is referenced by:  fnunop  6597  brwdom2  9459  sseqfn  34398  bnj927  34776  ofun  42268  tfsconcatfn  43370