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Theorem fvun1 5606
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 5357 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 976 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5357 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 977 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5359 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5359 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3385 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2304 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 214 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 454 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1148 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 fvun 5605 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  X
)  =  ( ( F `  X )  u.  ( G `  X ) ) )
132, 4, 11, 12syl21anc 1181 . 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 3514 . . . . . . . 8  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  A )  ->  -.  X  e.  B )
1514adantl 452 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  B
)
166eleq2d 2363 . . . . . . . 8  |-  ( G  Fn  B  ->  ( X  e.  dom  G  <->  X  e.  B ) )
1716adantr 451 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  G  <-> 
X  e.  B ) )
1815, 17mtbird 292 . . . . . 6  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
19183adant1 973 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
20 ndmfv 5568 . . . . 5  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
2119, 20syl 15 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( G `  X
)  =  (/) )
2221uneq2d 3342 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( ( F `  X )  u.  (/) ) )
23 un0 3492 . . 3  |-  ( ( F `  X )  u.  (/) )  =  ( F `  X )
2422, 23syl6eq 2344 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( F `
 X ) )
2513, 24eqtrd 2328 1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( F `
 X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163    i^i cin 3164   (/)c0 3468   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  fvun2  5607  hashf1lem1  11409  xpsc0  13478  ptunhmeo  17515  isoun  23257  cvmliftlem5  23835  vdgrun  23908  fullfunfv  24557  axlowdimlem6  24647  axlowdimlem8  24649  axlowdimlem11  24652  enfixsn  27360  constr3lem4  28393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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