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Theorem f1oun 5492
Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
Assertion
Ref Expression
f1oun  |-  ( ( ( F : A -1-1-onto-> B  /\  G : C -1-1-onto-> D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( F  u.  G ) : ( A  u.  C ) -1-1-onto-> ( B  u.  D ) )

Proof of Theorem f1oun
StepHypRef Expression
1 dff1o4 5480 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
2 dff1o4 5480 . . . 4  |-  ( G : C -1-1-onto-> D  <->  ( G  Fn  C  /\  `' G  Fn  D ) )
3 fnun 5350 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
43ex 423 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  C )  ->  ( ( A  i^i  C )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  C
) ) )
5 fnun 5350 . . . . . . . 8  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
6 cnvun 5086 . . . . . . . . 9  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
76fneq1i 5338 . . . . . . . 8  |-  ( `' ( F  u.  G
)  Fn  ( B  u.  D )  <->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
85, 7sylibr 203 . . . . . . 7  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  `' ( F  u.  G )  Fn  ( B  u.  D ) )
98ex 423 . . . . . 6  |-  ( ( `' F  Fn  B  /\  `' G  Fn  D
)  ->  ( ( B  i^i  D )  =  (/)  ->  `' ( F  u.  G )  Fn  ( B  u.  D
) ) )
104, 9im2anan9 808 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( `' F  Fn  B  /\  `' G  Fn  D
) )  ->  (
( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  ->  ( ( F  u.  G )  Fn  ( A  u.  C )  /\  `' ( F  u.  G
)  Fn  ( B  u.  D ) ) ) )
1110an4s 799 . . . 4  |-  ( ( ( F  Fn  A  /\  `' F  Fn  B
)  /\  ( G  Fn  C  /\  `' G  Fn  D ) )  -> 
( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  -> 
( ( F  u.  G )  Fn  ( A  u.  C )  /\  `' ( F  u.  G )  Fn  ( B  u.  D )
) ) )
121, 2, 11syl2anb 465 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  G : C -1-1-onto-> D )  ->  (
( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  ->  ( ( F  u.  G )  Fn  ( A  u.  C )  /\  `' ( F  u.  G
)  Fn  ( B  u.  D ) ) ) )
13 dff1o4 5480 . . 3  |-  ( ( F  u.  G ) : ( A  u.  C ) -1-1-onto-> ( B  u.  D
)  <->  ( ( F  u.  G )  Fn  ( A  u.  C
)  /\  `' ( F  u.  G )  Fn  ( B  u.  D
) ) )
1412, 13syl6ibr 218 . 2  |-  ( ( F : A -1-1-onto-> B  /\  G : C -1-1-onto-> D )  ->  (
( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  C
)
-1-1-onto-> ( B  u.  D
) ) )
1514imp 418 1  |-  ( ( ( F : A -1-1-onto-> B  /\  G : C -1-1-onto-> D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( F  u.  G ) : ( A  u.  C ) -1-1-onto-> ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    u. cun 3150    i^i cin 3151   (/)c0 3455   `'ccnv 4688    Fn wfn 5250   -1-1-onto->wf1o 5254
This theorem is referenced by:  fveqf1o  5806  oacomf1o  6563  unen  6943  domss2  7020  isinf  7076  marypha1lem  7186  hashf1lem1  11393  isoun  23242  subfacp1lem2a  23711  subfacp1lem5  23715  eupap1  23900  eldioph2lem1  26839  eldioph2lem2  26840  enfixsn  27257  f1oprg  28075  f1oun2prg  28076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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