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Theorem fununfun 5557
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 5530 . . 3  |-  ( Fun  ( F  u.  G
)  ->  Rel  ( F  u.  G ) )
2 relun 5051 . . 3  |-  ( Rel  ( F  u.  G
)  <->  ( Rel  F  /\  Rel  G ) )
31, 2sylib 196 . 2  |-  ( Fun  ( F  u.  G
)  ->  ( Rel  F  /\  Rel  G ) )
4 simpl 457 . . . . 5  |-  ( ( Rel  F  /\  Rel  G )  ->  Rel  F )
5 fununmo 5556 . . . . . 6  |-  ( Fun  ( F  u.  G
)  ->  E* y  x F y )
65alrimiv 1686 . . . . 5  |-  ( Fun  ( F  u.  G
)  ->  A. x E* y  x F
y )
74, 6anim12i 566 . . . 4  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  ( Rel  F  /\  A. x E* y  x F
y ) )
8 dffun6 5528 . . . 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 461 . . . . 5  |-  ( ( Rel  F  /\  Rel  G )  ->  Rel  G )
11 uncom 3595 . . . . . . . 8  |-  ( F  u.  G )  =  ( G  u.  F
)
1211funeqi 5533 . . . . . . 7  |-  ( Fun  ( F  u.  G
)  <->  Fun  ( G  u.  F ) )
13 fununmo 5556 . . . . . . 7  |-  ( Fun  ( G  u.  F
)  ->  E* y  x G y )
1412, 13sylbi 195 . . . . . 6  |-  ( Fun  ( F  u.  G
)  ->  E* y  x G y )
1514alrimiv 1686 . . . . 5  |-  ( Fun  ( F  u.  G
)  ->  A. x E* y  x G
y )
1610, 15anim12i 566 . . . 4  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  ( Rel  G  /\  A. x E* y  x G
y ) )
17 dffun6 5528 . . . 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 532 . 2  |-  ( ( ( Rel  F  /\  Rel  G )  /\  Fun  ( F  u.  G
) )  ->  ( Fun  F  /\  Fun  G
) )
203, 19mpancom 669 1  |-  ( Fun  ( F  u.  G
)  ->  ( Fun  F  /\  Fun  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368   E*wmo 2261    u. cun 3421   class class class wbr 4387   Rel wrel 4940   Fun wfun 5507
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-fun 5515
This theorem is referenced by:  fsuppunbi  7739
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