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Theorem fununfun 5614
Description: If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
Assertion
Ref Expression
fununfun  |-  ( Fun  ( F  u.  G
)  ->  ( Fun  F  /\  Fun  G ) )

Proof of Theorem fununfun
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funrel 5587 . . 3  |-  ( Fun  ( F  u.  G
)  ->  Rel  ( F  u.  G ) )
2 relun 5107 . . 3  |-  ( Rel  ( F  u.  G
)  <->  ( Rel  F  /\  Rel  G ) )
31, 2sylib 196 . 2  |-  ( Fun  ( F  u.  G
)  ->  ( Rel  F  /\  Rel  G ) )
4 simpl 455 . . . . 5  |-  ( ( Rel  F  /\  Rel  G )  ->  Rel  F )
5 fununmo 5613 . . . . . 6  |-  ( Fun  ( F  u.  G
)  ->  E* y  x F y )
65alrimiv 1724 . . . . 5  |-  ( Fun  ( F  u.  G
)  ->  A. x E* y  x F
y )
74, 6anim12i 564 . . . 4  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  ( Rel  F  /\  A. x E* y  x F
y ) )
8 dffun6 5585 . . . 4  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x E* y  x F y ) )
97, 8sylibr 212 . . 3  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  Fun  F )
10 simpr 459 . . . . 5  |-  ( ( Rel  F  /\  Rel  G )  ->  Rel  G )
11 uncom 3634 . . . . . . . 8  |-  ( F  u.  G )  =  ( G  u.  F
)
1211funeqi 5590 . . . . . . 7  |-  ( Fun  ( F  u.  G
)  <->  Fun  ( G  u.  F ) )
13 fununmo 5613 . . . . . . 7  |-  ( Fun  ( G  u.  F
)  ->  E* y  x G y )
1412, 13sylbi 195 . . . . . 6  |-  ( Fun  ( F  u.  G
)  ->  E* y  x G y )
1514alrimiv 1724 . . . . 5  |-  ( Fun  ( F  u.  G
)  ->  A. x E* y  x G
y )
1610, 15anim12i 564 . . . 4  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  ( Rel  G  /\  A. x E* y  x G
y ) )
17 dffun6 5585 . . . 4  |-  ( Fun 
G  <->  ( Rel  G  /\  A. x E* y  x G y ) )
1816, 17sylibr 212 . . 3  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  Fun  G )
199, 18jca 530 . 2  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  ( Fun  F  /\  Fun  G
) )
203, 19mpancom 667 1  |-  ( Fun  ( F  u.  G
)  ->  ( Fun  F  /\  Fun  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396   E*wmo 2285    u. cun 3459   class class class wbr 4439   Rel wrel 4993   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-fun 5572
This theorem is referenced by:  fsuppunbi  7842
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