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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 5335 . . 3  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 1020 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5335 . . 3  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 1021 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5337 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5337 . . . . . . 7  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3353 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2198 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 158 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 279 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1202 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 simp3r 1028 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  A )
135eleq2d 2259 . . . 4  |-  ( F  Fn  A  ->  ( X  e.  dom  F  <->  X  e.  A ) )
14133ad2ant1 1020 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  F  <-> 
X  e.  A ) )
1512, 14mpbird 167 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  dom  F )
16 funun 5282 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
17 ssun1 3313 . . . . . . . . 9  |-  F  C_  ( F  u.  G
)
18 dmss 4847 . . . . . . . . 9  |-  ( F 
C_  ( F  u.  G )  ->  dom  F 
C_  dom  ( F  u.  G ) )
1917, 18ax-mp 5 . . . . . . . 8  |-  dom  F  C_ 
dom  ( F  u.  G )
2019sseli 3166 . . . . . . 7  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  u.  G ) )
2116, 20anim12i 338 . . . . . 6  |-  ( ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G )  =  (/) )  /\  X  e.  dom  F )  ->  ( Fun  ( F  u.  G )  /\  X  e.  dom  ( F  u.  G
) ) )
2221anasss 399 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G )  /\  (
( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
) )  ->  ( Fun  ( F  u.  G
)  /\  X  e.  dom  ( F  u.  G
) ) )
23223impa 1196 . . . 4  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( Fun  ( F  u.  G )  /\  X  e.  dom  ( F  u.  G
) ) )
24 funfvdm 5603 . . . 4  |-  ( ( Fun  ( F  u.  G )  /\  X  e.  dom  ( F  u.  G ) )  -> 
( ( F  u.  G ) `  X
)  =  U. (
( F  u.  G
) " { X } ) )
2523, 24syl 14 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F  u.  G ) `  X )  =  U. ( ( F  u.  G ) " { X } ) )
26 imaundir 5063 . . . . . 6  |-  ( ( F  u.  G )
" { X }
)  =  ( ( F " { X } )  u.  ( G " { X }
) )
2726a1i 9 . . . . 5  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F  u.  G ) " { X } )  =  ( ( F " { X } )  u.  ( G " { X } ) ) )
2827unieqd 3838 . . . 4  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( ( F  u.  G ) " { X } )  = 
U. ( ( F
" { X }
)  u.  ( G
" { X }
) ) )
29 disjel 3492 . . . . . . . . 9  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  -.  X  e.  dom  G )
30 ndmima 5026 . . . . . . . . 9  |-  ( -.  X  e.  dom  G  ->  ( G " { X } )  =  (/) )
3129, 30syl 14 . . . . . . . 8  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  ( G " { X } )  =  (/) )
32313ad2ant3 1022 . . . . . . 7  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( G " { X } )  =  (/) )
3332uneq2d 3304 . . . . . 6  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  ( ( F " { X } )  u.  (/) ) )
34 un0 3471 . . . . . 6  |-  ( ( F " { X } )  u.  (/) )  =  ( F " { X } )
3533, 34eqtrdi 2238 . . . . 5  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  ( F " { X } ) )
3635unieqd 3838 . . . 4  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  U. ( F " { X } ) )
3728, 36eqtrd 2222 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( ( F  u.  G ) " { X } )  = 
U. ( F " { X } ) )
38 funfvdm 5603 . . . . . 6  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F `  X
)  =  U. ( F " { X }
) )
3938eqcomd 2195 . . . . 5  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  U. ( F " { X } )  =  ( F `  X ) )
4039adantrl 478 . . . 4  |-  ( ( Fun  F  /\  (
( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
) )  ->  U. ( F " { X }
)  =  ( F `
 X ) )
41403adant2 1018 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( F " { X } )  =  ( F `  X
) )
4225, 37, 413eqtrd 2226 . 2  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F  u.  G ) `  X )  =  ( F `  X ) )
432, 4, 11, 15, 42syl112anc 1253 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    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160    u. cun 3142    i^i cin 3143    C_ wss 3144   (/)c0 3437   {csn 3610   U.cuni 3827   dom cdm 4647   "cima 4650   Fun wfun 5232    Fn wfn 5233   ` cfv 5238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246
This theorem is referenced by:  fvun2  5607  caseinl  7124
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