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Theorem fvun2 5782
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 3519 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21fveq1i 5711 . 2  |-  ( ( F  u.  G ) `
 X )  =  ( ( G  u.  F ) `  X
)
3 incom 3562 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43eqeq1i 2450 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
54anbi1i 695 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  B )  <->  ( ( B  i^i  A )  =  (/)  /\  X  e.  B
) )
6 fvun1 5781 . . . 4  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( B  i^i  A )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
75, 6syl3an3b 1256 . . 3  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
873com12 1191 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
92, 8syl5eq 2487 1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( F  u.  G ) `  X
)  =  ( G `
 X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3345    i^i cin 3346   (/)c0 3656    Fn wfn 5432   ` cfv 5437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-fv 5445
This theorem is referenced by:  fveqf1o  6019  xpsc1  14518  ptunhmeo  19400  axlowdimlem9  23215  axlowdimlem12  23218  axlowdimlem17  23223  constr3lem4  23552  vdgrun  23590  vdgrfiun  23591  isoun  26016  resf1o  26049  sseqfv2  26796  cvmliftlem4  27196  fullfunfv  27997  finixpnum  28437
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