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Theorem fvun2 5921
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 3587 . . 3  |-  ( F  u.  G )  =  ( G  u.  F
)
21fveq1i 5850 . 2  |-  ( ( F  u.  G ) `
 X )  =  ( ( G  u.  F ) `  X
)
3 incom 3632 . . . . . 6  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43eqeq1i 2409 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  <->  ( B  i^i  A )  =  (/) )
54anbi1i 693 . . . 4  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  B )  <->  ( ( B  i^i  A )  =  (/)  /\  X  e.  B
) )
6 fvun1 5920 . . . 4  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( B  i^i  A )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
75, 6syl3an3b 1268 . . 3  |-  ( ( G  Fn  B  /\  F  Fn  A  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
873com12 1201 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  B ) )  -> 
( ( G  u.  F ) `  X
)  =  ( G `
 X ) )
92, 8syl5eq 2455 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    u. cun 3412    i^i cin 3413   (/)c0 3738    Fn wfn 5564   ` cfv 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-fv 5577
This theorem is referenced by:  fveqf1o  6188  xpsc1  15175  ptunhmeo  20601  axlowdimlem9  24670  axlowdimlem12  24673  axlowdimlem17  24678  constr3lem4  25064  vdgrun  25318  vdgrfiun  25319  isoun  27964  resf1o  28000  sseqfv2  28839  cvmliftlem4  29585  fullfunfv  30285  finixpnum  31410
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