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Theorem fvun1 5919
Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
Assertion
Ref Expression
fvun1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( F `
 X ) )

Proof of Theorem fvun1
StepHypRef Expression
1 fnfun 5660 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 1015 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5660 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 1016 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5662 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5662 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3688 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2456 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 223 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 466 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1191 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 fvun 5918 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  X
)  =  ( ( F `  X )  u.  ( G `  X ) ) )
132, 4, 11, 12syl21anc 1225 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( ( F `  X )  u.  ( G `  X ) ) )
14 disjel 3861 . . . . . . . 8  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  A )  ->  -.  X  e.  B )
1514adantl 464 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  B
)
166eleq2d 2524 . . . . . . . 8  |-  ( G  Fn  B  ->  ( X  e.  dom  G  <->  X  e.  B ) )
1716adantr 463 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  G  <-> 
X  e.  B ) )
1815, 17mtbird 299 . . . . . 6  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
19183adant1 1012 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
20 ndmfv 5872 . . . . 5  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
2119, 20syl 16 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( G `  X
)  =  (/) )
2221uneq2d 3644 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( ( F `  X )  u.  (/) ) )
23 un0 3809 . . 3  |-  ( ( F `  X )  u.  (/) )  =  ( F `  X )
2422, 23syl6eq 2511 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( F `
 X ) )
2513, 24eqtrd 2495 1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( F `
 X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    u. cun 3459    i^i cin 3460   (/)c0 3783   dom cdm 4988   Fun wfun 5564    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  fvun2  5920  enfixsn  7619  hashf1lem1  12491  xpsc0  15052  ptunhmeo  20478  axlowdimlem6  24455  axlowdimlem8  24457  axlowdimlem11  24460  constr3lem4  24852  vdgrun  25106  vdgrfiun  25107  isoun  27751  sseqfv1  28595  cvmliftlem5  29001  fullfunfv  29828  finixpnum  30281  aacllem  33623
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