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Theorem fun2 5406
Description: The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
fun2  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> C )

Proof of Theorem fun2
StepHypRef Expression
1 fun 5405 . 2  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> ( C  u.  C ) )
2 unidm 3318 . . 3  |-  ( C  u.  C )  =  C
3 feq3 5377 . . 3  |-  ( ( C  u.  C )  =  C  ->  (
( F  u.  G
) : ( A  u.  B ) --> ( C  u.  C )  <-> 
( F  u.  G
) : ( A  u.  B ) --> C ) )
42, 3ax-mp 8 . 2  |-  ( ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  C )  <->  ( F  u.  G ) : ( A  u.  B ) --> C )
51, 4sylib 188 1  |-  ( ( ( F : A --> C  /\  G : B --> C )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B
) --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    u. cun 3150    i^i cin 3151   (/)c0 3455   -->wf 5251
This theorem is referenced by:  fresaun  5412  mapunen  7030  ac6sfi  7101  axdc3lem4  8079  fseq1p1m1  10857  umgraun  23879  eupap1  23900  axlowdimlem5  24574  axlowdimlem7  24576  clscnc  26010  uslgraun  28120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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