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Theorem fvun1 5874
Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
Assertion
Ref Expression
fvun1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( F `
 X ) )

Proof of Theorem fvun1
StepHypRef Expression
1 fnfun 5619 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
213ad2ant1 1009 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  F )
3 fnfun 5619 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
433ad2ant2 1010 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  Fun  G )
5 fndm 5621 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
6 fndm 5621 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
75, 6ineqan12d 3665 . . . . . . 7  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
87eqeq1d 2456 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( dom  F  i^i  dom  G )  =  (/) 
<->  ( A  i^i  B
)  =  (/) ) )
98biimprd 223 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( A  i^i  B )  =  (/)  ->  ( dom  F  i^i  dom  G
)  =  (/) ) )
109adantrd 468 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( ( A  i^i  B )  =  (/)  /\  X  e.  A
)  ->  ( dom  F  i^i  dom  G )  =  (/) ) )
11103impia 1185 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( dom  F  i^i  dom 
G )  =  (/) )
12 fvun 5873 . . 3  |-  ( ( ( Fun  F  /\  Fun  G )  /\  ( dom  F  i^i  dom  G
)  =  (/) )  -> 
( ( F  u.  G ) `  X
)  =  ( ( F `  X )  u.  ( G `  X ) ) )
132, 4, 11, 12syl21anc 1218 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( ( F `  X )  u.  ( G `  X ) ) )
14 disjel 3836 . . . . . . . 8  |-  ( ( ( A  i^i  B
)  =  (/)  /\  X  e.  A )  ->  -.  X  e.  B )
1514adantl 466 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  B
)
166eleq2d 2524 . . . . . . . 8  |-  ( G  Fn  B  ->  ( X  e.  dom  G  <->  X  e.  B ) )
1716adantr 465 . . . . . . 7  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( X  e.  dom  G  <-> 
X  e.  B ) )
1815, 17mtbird 301 . . . . . 6  |-  ( ( G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
19183adant1 1006 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  ->  -.  X  e.  dom  G )
20 ndmfv 5826 . . . . 5  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
2119, 20syl 16 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( G `  X
)  =  (/) )
2221uneq2d 3621 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( ( F `  X )  u.  (/) ) )
23 un0 3773 . . 3  |-  ( ( F `  X )  u.  (/) )  =  ( F `  X )
2422, 23syl6eq 2511 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F `  X )  u.  ( G `  X )
)  =  ( F `
 X ) )
2513, 24eqtrd 2495 1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( ( A  i^i  B )  =  (/)  /\  X  e.  A ) )  -> 
( ( F  u.  G ) `  X
)  =  ( F `
 X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    u. cun 3437    i^i cin 3438   (/)c0 3748   dom cdm 4951   Fun wfun 5523    Fn wfn 5524   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-fv 5537
This theorem is referenced by:  fvun2  5875  enfixsn  7533  hashf1lem1  12329  xpsc0  14620  ptunhmeo  19516  axlowdimlem6  23365  axlowdimlem8  23367  axlowdimlem11  23370  constr3lem4  23705  vdgrun  23743  vdgrfiun  23744  isoun  26168  sseqfv1  26936  cvmliftlem5  27342  fullfunfv  28142  finixpnum  28582
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