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Theorem foun 5461
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 5422 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5422 . . . 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 5304 . . 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 5019 . . 3  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
7 forn 5423 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87ad2antrr 485 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  F  =  B )
9 forn 5423 . . . . 5  |-  ( G : C -onto-> D  ->  ran  G  =  D )
109ad2antlr 486 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  G  =  D )
118, 10uneq12d 3282 . . 3  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( ran 
F  u.  ran  G
)  =  ( B  u.  D ) )
126, 11eqtrid 2215 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  ( F  u.  G )  =  ( B  u.  D ) )
13 df-fo 5204 . 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 415 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 1348    u. cun 3119    i^i cin 3120   (/)c0 3414   ran crn 4612    Fn wfn 5193   -onto->wfo 5196
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204
This theorem is referenced by: (None)
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