Theorem fun2d 6772

Description: The union of functions with disjoint domains is a function, deduction version of fun2 6771. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
fun2d.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
fun2d.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
fun2d.i	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
fun2d	⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)

Proof of Theorem fun2d

Step	Hyp	Ref	Expression
1		fun2d.f	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
2		fun2d.g	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
3		fun2d.i	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
4		fun2 6771	. 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
5	1, 2, 3, 4	syl21anc 838	1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)

Syntax hints:   → wi 4   = wceq 1540   ∪ cun 3949   ∩ cin 3950  ∅c0 4333  ⟶wf 6557

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-rn 5696  df-fun 6563  df-fn 6564  df-f 6565

This theorem is referenced by:  fresaun  6779  mapunen  9186  ac6sfi  9320  axdc3lem4  10493  fseq1p1m1  13638  uhgrun  29091  upgrun  29135  umgrun  29137  elrspunidl  33456  lbsdiflsp0  33677  reprsuc  34630  dvun  42389  evlselvlem  42596  evlselv  42597