Theorem fun2d 6773

Description: The union of functions with disjoint domains is a function, deduction version of fun2 6772. (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 6772	. 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 1537   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by:  fresaun  6780  mapunen  9185  ac6sfi  9318  axdc3lem4  10491  fseq1p1m1  13635  uhgrun  29106  upgrun  29150  umgrun  29152  elrspunidl  33436  lbsdiflsp0  33654  reprsuc  34609  dvun  42368  evlselvlem  42573  evlselv  42574