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Theorem foun 5696
Description: The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
foun  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  C
) -onto-> ( B  u.  D ) )

Proof of Theorem foun
StepHypRef Expression
1 fofn 5658 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5658 . . . 4  |-  ( G : C -onto-> D  ->  G  Fn  C )
31, 2anim12i 551 . . 3  |-  ( ( F : A -onto-> B  /\  G : C -onto-> D
)  ->  ( F  Fn  A  /\  G  Fn  C ) )
4 fnun 5554 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
53, 4sylan 459 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C
) )
6 rnun 5283 . . 3  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
7 forn 5659 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87ad2antrr 708 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  F  =  B )
9 forn 5659 . . . . 5  |-  ( G : C -onto-> D  ->  ran  G  =  D )
109ad2antlr 709 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  G  =  D )
118, 10uneq12d 3504 . . 3  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( ran 
F  u.  ran  G
)  =  ( B  u.  D ) )
126, 11syl5eq 2482 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  ( F  u.  G )  =  ( B  u.  D ) )
13 df-fo 5463 . 2  |-  ( ( F  u.  G ) : ( A  u.  C ) -onto-> ( B  u.  D )  <->  ( ( F  u.  G )  Fn  ( A  u.  C
)  /\  ran  ( F  u.  G )  =  ( B  u.  D
) ) )
145, 12, 13sylanbrc 647 1  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  C
) -onto-> ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    u. cun 3320    i^i cin 3321   (/)c0 3630   ran crn 4882    Fn wfn 5452   -onto->wfo 5455
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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  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-rab 2716  df-v 2960  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-br 4216  df-opab 4270  df-id 4501  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463
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