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Theorem fvun1 5590
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 5341 . . . 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 5341 . . . 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 5343 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5343 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3372 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2291 . . . . . 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 5589 . . 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 3501 . . . . . . . 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 2350 . . . . . . . 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 5552 . . . . 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 3329 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( ( F `  X )  u.  (/) ) )
23 un0 3479 . . 3  |-  ( ( F `  X )  u.  (/) )  =  ( F `  X )
2422, 23syl6eq 2331 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( F `
 X ) )
2513, 24eqtrd 2315 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 1623    e. wcel 1684    u. cun 3150    i^i cin 3151   (/)c0 3455   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  fvun2  5591  hashf1lem1  11393  xpsc0  13462  ptunhmeo  17499  isoun  23242  cvmliftlem5  23820  vdgrun  23893  fullfunfv  24485  axlowdimlem6  24575  axlowdimlem8  24577  axlowdimlem11  24580  enfixsn  27257
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-13 1686  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-pow 4188  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-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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