ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  foun Unicode version

Theorem foun 5354
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 5317 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5317 . . . 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 5199 . . 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 4917 . . 3  |-  ran  ( F  u.  G )  =  ( ran  F  u.  ran  G )
7 forn 5318 . . . . 5  |-  ( F : A -onto-> B  ->  ran  F  =  B )
87ad2antrr 479 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  F  =  B )
9 forn 5318 . . . . 5  |-  ( G : C -onto-> D  ->  ran  G  =  D )
109ad2antlr 480 . . . 4  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  G  =  D )
118, 10uneq12d 3201 . . 3  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ( ran 
F  u.  ran  G
)  =  ( B  u.  D ) )
126, 11syl5eq 2162 . 2  |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D )  /\  ( A  i^i  C )  =  (/) )  ->  ran  ( F  u.  G )  =  ( B  u.  D ) )
13 df-fo 5099 . 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 413 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 1316    u. cun 3039    i^i cin 3040   (/)c0 3333   ran crn 4510    Fn wfn 5088   -onto->wfo 5091
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator