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Theorem fun2d 6227
Description: The union of functions with disjoint domains is a function, deduction version of fun2 6226. (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 6226 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 2, 3, 4syl21anc 1476 1 (𝜑 → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  cun 3711  cin 3712  c0 4056  wf 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-id 5172  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-fun 6049  df-fn 6050  df-f 6051
This theorem is referenced by:  uhgrun  26166  upgrun  26210  umgrun  26212  reprsuc  31000
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