Theorem fun2d 6698

Description: The union of functions with disjoint domains is a function, deduction version of fun2 6697. (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 6697	. 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 1542   ∪ cun 3888   ∩ cin 3889  ∅c0 4274  ⟶wf 6488

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  fresaun  6705  mapunen  9077  ac6sfi  9187  axdc3lem4  10366  fseq1p1m1  13543  uhgrun  29157  upgrun  29201  umgrun  29203  elrspunidl  33503  evlextv  33701  lbsdiflsp0  33786  reprsuc  34775  dvun  42805  evlselvlem  43033  evlselv  43034