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Theorem fvun2 5454
Description: The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
Assertion
Ref Expression
fvun2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( F  u.  G ) `  X
)  =  ( G `
 X ) )

Proof of Theorem fvun2
StepHypRef Expression
1 uncom 3188 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21fveq1i 5388 . 2  |-  ( ( F  u.  G ) `
 X )  =  ( ( G  u.  F ) `  X
)
3 incom 3236 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43eqeq1i 2123 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
54anbi1i 451 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  B )  <->  ( ( B  i^i  A )  =  (/)  /\  X  e.  B
) )
6 fvun1 5453 . . . 4  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( B  i^i  A )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
75, 6syl3an3b 1237 . . 3  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
873com12 1168 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
92, 8syl5eq 2160 1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( F  u.  G ) `  X
)  =  ( G `
 X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463    u. cun 3037    i^i cin 3038   (/)c0 3331    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  caseinr  6943
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