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Theorem fvun1 5267
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 5024 . . 3  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 936 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5024 . . 3  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 937 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5026 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5026 . . . . . . 7  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3168 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2064 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 151 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 268 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1112 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 simp3r 944 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  A )
135eleq2d 2123 . . . 4  |-  ( F  Fn  A  ->  ( X  e.  dom  F  <->  X  e.  A ) )
14133ad2ant1 936 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  F  <-> 
X  e.  A ) )
1512, 14mpbird 160 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  X  e.  dom  F )
16 funun 4972 . . . . . . 7  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  ->  Fun  ( F  u.  G
) )
17 ssun1 3134 . . . . . . . . 9  |-  F  C_  ( F  u.  G
)
18 dmss 4562 . . . . . . . . 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 2969 . . . . . . 7  |-  ( X  e.  dom  F  ->  X  e.  dom  ( F  u.  G ) )
2116, 20anim12i 325 . . . . . 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 385 . . . . 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 1110 . . . 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 5264 . . . 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 4765 . . . . . 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 3619 . . . 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 3302 . . . . . . . . 9  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  -.  X  e.  dom  G )
30 ndmima 4730 . . . . . . . . 9  |-  ( -.  X  e.  dom  G  ->  ( G " { X } )  =  (/) )
3129, 30syl 14 . . . . . . . 8  |-  ( ( ( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
)  ->  ( G " { X } )  =  (/) )
32313ad2ant3 938 . . . . . . 7  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( G " { X } )  =  (/) )
3332uneq2d 3125 . . . . . 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 3279 . . . . . 6  |-  ( ( F " { X } )  u.  (/) )  =  ( F " { X } )
3533, 34syl6eq 2104 . . . . 5  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  ( ( F
" { X }
)  u.  ( G
" { X }
) )  =  ( F " { X } ) )
3635unieqd 3619 . . . 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 2088 . . 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 5264 . . . . . 6  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F `  X
)  =  U. ( F " { X }
) )
3938eqcomd 2061 . . . . 5  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  U. ( F " { X } )  =  ( F `  X ) )
4039adantrl 455 . . . 4  |-  ( ( Fun  F  /\  (
( dom  F  i^i  dom 
G )  =  (/)  /\  X  e.  dom  F
) )  ->  U. ( F " { X }
)  =  ( F `
 X ) )
41403adant2 934 . . 3  |-  ( ( Fun  F  /\  Fun  G  /\  ( ( dom 
F  i^i  dom  G )  =  (/)  /\  X  e. 
dom  F ) )  ->  U. ( F " { X } )  =  ( F `  X
) )
4225, 37, 413eqtrd 2092 . 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 1150 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409    u. cun 2943    i^i cin 2944    C_ wss 2945   (/)c0 3252   {csn 3403   U.cuni 3608   dom cdm 4373   "cima 4376   Fun wfun 4924    Fn wfn 4925   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by:  fvun2  5268
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