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Theorem fun2 6228
Description: The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
fun2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)

Proof of Theorem fun2
StepHypRef Expression
1 fun 6227 . 2 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶))
2 unidm 3899 . . 3 (𝐶𝐶) = 𝐶
3 feq3 6189 . . 3 ((𝐶𝐶) = 𝐶 → ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶))
42, 3ax-mp 5 . 2 ((𝐹𝐺):(𝐴𝐵)⟶(𝐶𝐶) ↔ (𝐹𝐺):(𝐴𝐵)⟶𝐶)
51, 4sylib 208 1 (((𝐹:𝐴𝐶𝐺:𝐵𝐶) ∧ (𝐴𝐵) = ∅) → (𝐹𝐺):(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  cun 3713  cin 3714  c0 4058  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-fun 6051  df-fn 6052  df-f 6053
This theorem is referenced by:  fun2d  6229  fresaun  6236  mapunen  8296  ac6sfi  8371  axdc3lem4  9487  fseq1p1m1  12627  axlowdimlem5  26046  axlowdimlem7  26048  resf1o  29835  locfinref  30238  breprexplema  31038
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