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Theorem fvun2 5591
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 3319 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21fveq1i 5526 . 2  |-  ( ( F  u.  G ) `
 X )  =  ( ( G  u.  F ) `  X
)
3 incom 3361 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43eqeq1i 2290 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
54anbi1i 676 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  B )  <->  ( ( B  i^i  A )  =  (/)  /\  X  e.  B
) )
6 fvun1 5590 . . . 4  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( B  i^i  A )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
75, 6syl3an3b 1220 . . 3  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
873com12 1155 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
92, 8syl5eq 2327 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150    i^i cin 3151   (/)c0 3455    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  fveqf1o  5806  xpsc1  13463  ptunhmeo  17499  isoun  23242  cvmliftlem4  23819  vdgrun  23893  fullfunfv  24485  axlowdimlem9  24578  axlowdimlem12  24581  axlowdimlem17  24586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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