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Theorem f1oun 5696
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 5684 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
2 dff1o4 5684 . . . 4  |-  ( G : C -1-1-onto-> D  <->  ( G  Fn  C  /\  `' G  Fn  D ) )
3 fnun 5553 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
43ex 425 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  C )  ->  ( ( A  i^i  C )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  C
) ) )
5 fnun 5553 . . . . . . . 8  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
6 cnvun 5279 . . . . . . . . 9  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
76fneq1i 5541 . . . . . . . 8  |-  ( `' ( F  u.  G
)  Fn  ( B  u.  D )  <->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
85, 7sylibr 205 . . . . . . 7  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  `' ( F  u.  G )  Fn  ( B  u.  D ) )
98ex 425 . . . . . 6  |-  ( ( `' F  Fn  B  /\  `' G  Fn  D
)  ->  ( ( B  i^i  D )  =  (/)  ->  `' ( F  u.  G )  Fn  ( B  u.  D
) ) )
104, 9im2anan9 810 . . . . 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 801 . . . 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 467 . . 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 5684 . . 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 220 . 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 420 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 360    = wceq 1653    u. cun 3320    i^i cin 3321   (/)c0 3630   `'ccnv 4879    Fn wfn 5451   -1-1-onto->wf1o 5455
This theorem is referenced by:  f1oprg  5720  fveqf1o  6031  oacomf1o  6810  unen  7191  domss2  7268  isinf  7324  marypha1lem  7440  hashf1lem1  11706  f1oun2prg  11866  eupap1  21700  isoun  24091  subfacp1lem2a  24868  subfacp1lem5  24872  eldioph2lem1  26820  eldioph2lem2  26821  enfixsn  27236
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-id 4500  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463
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