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 843	1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)

Syntax hints:   → wi 4   = wceq 1547   ∪ cun 3888   ∩ cin 3889  ∅c0 4268  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  fresaun  6705  mapunen  9081  ac6sfi  9191  axdc3lem4  10373  fseq1p1m1  13550  uhgrun  29168  upgrun  29212  umgrun  29214  elrspunidl  33518  evlextv  33733  lbsdiflsp0  33817  reprsuc  34806  dvun  42843  evlselvlem  43045  evlselv  43046