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Theorem foun 5433
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 5394 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5394 . . . 4  |-  ( G : C -onto-> D  ->  G  Fn  C )
31, 2anim12i 336 . . 3  |-  ( ( F : A -onto-> B  /\  G : C -onto-> D
)  ->  ( F  Fn  A  /\  G  Fn  C ) )
4 fnun 5276 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
53, 4sylan 281 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C
) )
6 rnun 4994 . . 3  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
7 forn 5395 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87ad2antrr 480 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  F  =  B )
9 forn 5395 . . . . 5  |-  ( G : C -onto-> D  ->  ran  G  =  D )
109ad2antlr 481 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  G  =  D )
118, 10uneq12d 3262 . . 3  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( ran 
F  u.  ran  G
)  =  ( B  u.  D ) )
126, 11syl5eq 2202 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  ( F  u.  G )  =  ( B  u.  D ) )
13 df-fo 5176 . 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 414 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 103    = wceq 1335    u. cun 3100    i^i cin 3101   (/)c0 3394   ran crn 4587    Fn wfn 5165   -onto->wfo 5168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-fun 5172  df-fn 5173  df-f 5174  df-fo 5176
This theorem is referenced by: (None)
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