Theorem fun2d 6749

Description: The union of functions with disjoint domains is a function, deduction version of fun2 6748. (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 6748	. 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
5	1, 2, 3, 4	syl21anc 851	1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∪ cun 3906   ∩ cin 3907  ∅c0 4289  ⟶wf 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6545  df-fn 6546  df-f 6547

This theorem is used by:  fresaun  6756  mapunen  9144  ac6sfi  9254  axdc3lem4  10455  fseq1p1m1  13645  uhgrun  29461  upgrun  29505  umgrun  29507  elrspunidl  33767  evlextv  33963  lbsdiflsp0  34047  reprsuc  35034  dvun  43161  evlselvlem  43361  evlselv  43362