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Theorem fun2d 5981
 Description: The union of functions with disjoint domains is a function, deduction version of fun2 5980. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
Hypotheses
Ref Expression
fun2d.f (𝜑𝐹:𝐴𝐶)
fun2d.g (𝜑𝐺:𝐵𝐶)
fun2d.i (𝜑 → (𝐴𝐵) = ∅)
Assertion
Ref Expression
fun2d (𝜑 → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Proof of Theorem fun2d
StepHypRef Expression
1 fun2d.f . 2 (𝜑𝐹:𝐴𝐶)
2 fun2d.g . 2 (𝜑𝐺:𝐵𝐶)
3 fun2d.i . 2 (𝜑 → (𝐴𝐵) = ∅)
4 fun2 5980 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl21anc 1317 1 (𝜑 → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  uhgrun  25740  upgrun  25784  umgrun  25786
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