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Theorem fvun1 5239
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 4996 . . 3  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 925 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 4996 . . 3  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 926 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 4998 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 4998 . . . . . . 7  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3140 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2048 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 147 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 264 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1101 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 simp3r 933 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  A )
135eleq2d 2107 . . . 4  |-  ( F  Fn  A  ->  ( X  e.  dom  F  <->  X  e.  A ) )
14133ad2ant1 925 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  F  <-> 
X  e.  A ) )
1512, 14mpbird 156 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  dom  F )
16 funun 4944 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
17 ssun1 3106 . . . . . . . . 9  |-  F  C_  ( F  u.  G
)
18 dmss 4534 . . . . . . . . 9  |-  ( F 
C_  ( F  u.  G )  ->  dom  F 
C_  dom  ( F  u.  G ) )
1917, 18ax-mp 7 . . . . . . . 8  |-  dom  F  C_ 
dom  ( F  u.  G )
2019sseli 2941 . . . . . . 7  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  u.  G ) )
2116, 20anim12i 321 . . . . . 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 379 . . . . 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 1099 . . . 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 5236 . . . 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 4737 . . . . . 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 3591 . . . 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 3274 . . . . . . . . 9  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  -.  X  e.  dom  G )
30 ndmima 4702 . . . . . . . . 9  |-  ( -.  X  e.  dom  G  ->  ( G " { X } )  =  (/) )
3129, 30syl 14 . . . . . . . 8  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  ( G " { X } )  =  (/) )
32313ad2ant3 927 . . . . . . 7  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( G " { X } )  =  (/) )
3332uneq2d 3097 . . . . . 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 3251 . . . . . 6  |-  ( ( F " { X } )  u.  (/) )  =  ( F " { X } )
3533, 34syl6eq 2088 . . . . 5  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  ( F " { X } ) )
3635unieqd 3591 . . . 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 2072 . . 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 5236 . . . . . 6  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F `  X
)  =  U. ( F " { X }
) )
3938eqcomd 2045 . . . . 5  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  U. ( F " { X } )  =  ( F `  X ) )
4039adantrl 447 . . . 4  |-  ( ( Fun  F  /\  (
( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
) )  ->  U. ( F " { X }
)  =  ( F `
 X ) )
41403adant2 923 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( F " { X } )  =  ( F `  X
) )
4225, 37, 413eqtrd 2076 . 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 1139 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 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393    u. cun 2915    i^i cin 2916    C_ wss 2917   (/)c0 3224   {csn 3375   U.cuni 3580   dom cdm 4345   "cima 4348   Fun wfun 4896    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  fvun2  5240
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