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Theorem fun2d 6515
Description: The union of functions with disjoint domains is a function, deduction version of fun2 6514. (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 6514 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl21anc 836 1 (𝜑 → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cun 3908  cin 3909  c0 4266  wf 6324
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-id 5433  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-fun 6330  df-fn 6331  df-f 6332
This theorem is referenced by:  fresaun  6522  mapunen  8662  ac6sfi  8738  axdc3lem4  9852  fseq1p1m1  12964  uhgrun  26846  upgrun  26890  umgrun  26892  lbsdiflsp0  31033  reprsuc  31894
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