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Theorem foun 5145
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 5108 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5108 . . . 4  |-  ( G : C -onto-> D  ->  G  Fn  C )
31, 2anim12i 321 . . 3  |-  ( ( F : A -onto-> B  /\  G : C -onto-> D
)  ->  ( F  Fn  A  /\  G  Fn  C ) )
4 fnun 5005 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
53, 4sylan 267 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C
) )
6 rnun 4732 . . 3  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
7 forn 5109 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87ad2antrr 457 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  F  =  B )
9 forn 5109 . . . . 5  |-  ( G : C -onto-> D  ->  ran  G  =  D )
109ad2antlr 458 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  G  =  D )
118, 10uneq12d 3098 . . 3  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( ran 
F  u.  ran  G
)  =  ( B  u.  D ) )
126, 11syl5eq 2084 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  ( F  u.  G )  =  ( B  u.  D ) )
13 df-fo 4908 . 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 394 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 97    = wceq 1243    u. cun 2915    i^i cin 2916   (/)c0 3224   ran crn 4346    Fn wfn 4897   -onto->wfo 4900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908
This theorem is referenced by: (None)
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