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Theorem fvun1 5795
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 5543 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 979 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5543 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 980 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5545 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5545 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3545 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2445 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 216 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 456 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1151 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 fvun 5794 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  X
)  =  ( ( F `  X )  u.  ( G `  X ) ) )
132, 4, 11, 12syl21anc 1184 . 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 3675 . . . . . . . 8  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  A )  ->  -.  X  e.  B )
1514adantl 454 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  B
)
166eleq2d 2504 . . . . . . . 8  |-  ( G  Fn  B  ->  ( X  e.  dom  G  <->  X  e.  B ) )
1716adantr 453 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  G  <-> 
X  e.  B ) )
1815, 17mtbird 294 . . . . . 6  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
19183adant1 976 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
20 ndmfv 5756 . . . . 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 3502 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( ( F `  X )  u.  (/) ) )
23 un0 3653 . . 3  |-  ( ( F `  X )  u.  (/) )  =  ( F `  X )
2422, 23syl6eq 2485 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( F `
 X ) )
2513, 24eqtrd 2469 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3319    i^i cin 3320   (/)c0 3629   dom cdm 4879   Fun wfun 5449    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  fvun2  5796  hashf1lem1  11705  xpsc0  13786  ptunhmeo  17841  constr3lem4  21635  vdgrun  21673  vdgrfiun  21674  isoun  24090  cvmliftlem5  24977  fullfunfv  25793  axlowdimlem6  25887  axlowdimlem8  25889  axlowdimlem11  25892  enfixsn  27235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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