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Theorem fvun1 5370
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 5111 . . 3  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 964 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5111 . . 3  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 965 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5113 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5113 . . . . . . 7  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3203 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2096 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 156 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 273 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1140 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 simp3r 972 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  A )
135eleq2d 2157 . . . 4  |-  ( F  Fn  A  ->  ( X  e.  dom  F  <->  X  e.  A ) )
14133ad2ant1 964 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  F  <-> 
X  e.  A ) )
1512, 14mpbird 165 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  dom  F )
16 funun 5058 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
17 ssun1 3163 . . . . . . . . 9  |-  F  C_  ( F  u.  G
)
18 dmss 4635 . . . . . . . . 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 3021 . . . . . . 7  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  u.  G ) )
2116, 20anim12i 331 . . . . . 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 391 . . . . 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 1138 . . . 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 5367 . . . 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 4845 . . . . . 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 3664 . . . 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 3337 . . . . . . . . 9  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  -.  X  e.  dom  G )
30 ndmima 4809 . . . . . . . . 9  |-  ( -.  X  e.  dom  G  ->  ( G " { X } )  =  (/) )
3129, 30syl 14 . . . . . . . 8  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  ( G " { X } )  =  (/) )
32313ad2ant3 966 . . . . . . 7  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( G " { X } )  =  (/) )
3332uneq2d 3154 . . . . . 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 3316 . . . . . 6  |-  ( ( F " { X } )  u.  (/) )  =  ( F " { X } )
3533, 34syl6eq 2136 . . . . 5  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  ( F " { X } ) )
3635unieqd 3664 . . . 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 2120 . . 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 5367 . . . . . 6  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F `  X
)  =  U. ( F " { X }
) )
3938eqcomd 2093 . . . . 5  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  U. ( F " { X } )  =  ( F `  X ) )
4039adantrl 462 . . . 4  |-  ( ( Fun  F  /\  (
( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
) )  ->  U. ( F " { X }
)  =  ( F `
 X ) )
41403adant2 962 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( F " { X } )  =  ( F `  X
) )
4225, 37, 413eqtrd 2124 . 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 1178 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438    u. cun 2997    i^i cin 2998    C_ wss 2999   (/)c0 3286   {csn 3446   U.cuni 3653   dom cdm 4438   "cima 4441   Fun wfun 5009    Fn wfn 5010   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  fvun2  5371
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