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Theorem fvun1 5405
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 5145 . . 3  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 967 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5145 . . 3  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 968 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5147 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5147 . . . . . . 7  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3218 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2103 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 157 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 274 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1143 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 simp3r 975 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  A )
135eleq2d 2164 . . . 4  |-  ( F  Fn  A  ->  ( X  e.  dom  F  <->  X  e.  A ) )
14133ad2ant1 967 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  F  <-> 
X  e.  A ) )
1512, 14mpbird 166 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  dom  F )
16 funun 5092 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
17 ssun1 3178 . . . . . . . . 9  |-  F  C_  ( F  u.  G
)
18 dmss 4666 . . . . . . . . 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 3035 . . . . . . 7  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  u.  G ) )
2116, 20anim12i 332 . . . . . 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 392 . . . . 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 1141 . . . 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 5402 . . . 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 4878 . . . . . 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 3686 . . . 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 3356 . . . . . . . . 9  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  -.  X  e.  dom  G )
30 ndmima 4842 . . . . . . . . 9  |-  ( -.  X  e.  dom  G  ->  ( G " { X } )  =  (/) )
3129, 30syl 14 . . . . . . . 8  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  ( G " { X } )  =  (/) )
32313ad2ant3 969 . . . . . . 7  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( G " { X } )  =  (/) )
3332uneq2d 3169 . . . . . 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 3335 . . . . . 6  |-  ( ( F " { X } )  u.  (/) )  =  ( F " { X } )
3533, 34syl6eq 2143 . . . . 5  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  ( F " { X } ) )
3635unieqd 3686 . . . 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 2127 . . 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 5402 . . . . . 6  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F `  X
)  =  U. ( F " { X }
) )
3938eqcomd 2100 . . . . 5  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  U. ( F " { X } )  =  ( F `  X ) )
4039adantrl 463 . . . 4  |-  ( ( Fun  F  /\  (
( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
) )  ->  U. ( F " { X }
)  =  ( F `
 X ) )
41403adant2 965 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( F " { X } )  =  ( F `  X
) )
4225, 37, 413eqtrd 2131 . 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 1185 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 103    <-> wb 104    /\ w3a 927    = wceq 1296    e. wcel 1445    u. cun 3011    i^i cin 3012    C_ wss 3013   (/)c0 3302   {csn 3466   U.cuni 3675   dom cdm 4467   "cima 4470   Fun wfun 5043    Fn wfn 5044   ` cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057
This theorem is referenced by:  fvun2  5406  caseinl  6862
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