MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvun2 Structured version   Unicode version

Theorem fvun2 5798
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 3493 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21fveq1i 5732 . 2  |-  ( ( F  u.  G ) `
 X )  =  ( ( G  u.  F ) `  X
)
3 incom 3535 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43eqeq1i 2445 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
54anbi1i 678 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  B )  <->  ( ( B  i^i  A )  =  (/)  /\  X  e.  B
) )
6 fvun1 5797 . . . 4  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( B  i^i  A )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
75, 6syl3an3b 1223 . . 3  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
873com12 1158 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
92, 8syl5eq 2482 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3320    i^i cin 3321   (/)c0 3630    Fn wfn 5452   ` cfv 5457
This theorem is referenced by:  fveqf1o  6032  xpsc1  13791  ptunhmeo  17845  constr3lem4  21639  vdgrun  21677  vdgrfiun  21678  isoun  24094  cvmliftlem4  24980  fullfunfv  25797  axlowdimlem9  25894  axlowdimlem12  25897  axlowdimlem17  25902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
  Copyright terms: Public domain W3C validator