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Theorem f1oun 5653
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 5641 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
2 dff1o4 5641 . . . 4  |-  ( G : C -1-1-onto-> D  <->  ( G  Fn  C  /\  `' G  Fn  D ) )
3 fnun 5510 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
43ex 424 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  C )  ->  ( ( A  i^i  C )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  C
) ) )
5 fnun 5510 . . . . . . . 8  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
6 cnvun 5236 . . . . . . . . 9  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
76fneq1i 5498 . . . . . . . 8  |-  ( `' ( F  u.  G
)  Fn  ( B  u.  D )  <->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
85, 7sylibr 204 . . . . . . 7  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  `' ( F  u.  G )  Fn  ( B  u.  D ) )
98ex 424 . . . . . 6  |-  ( ( `' F  Fn  B  /\  `' G  Fn  D
)  ->  ( ( B  i^i  D )  =  (/)  ->  `' ( F  u.  G )  Fn  ( B  u.  D
) ) )
104, 9im2anan9 809 . . . . 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 800 . . . 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 466 . . 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 5641 . . 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 219 . 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 419 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 359    = wceq 1649    u. cun 3278    i^i cin 3279   (/)c0 3588   `'ccnv 4836    Fn wfn 5408   -1-1-onto->wf1o 5412
This theorem is referenced by:  f1oprg  5677  fveqf1o  5988  oacomf1o  6767  unen  7148  domss2  7225  isinf  7281  marypha1lem  7396  hashf1lem1  11659  f1oun2prg  11819  eupap1  21651  isoun  24042  subfacp1lem2a  24819  subfacp1lem5  24823  eldioph2lem1  26708  eldioph2lem2  26709  enfixsn  27125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
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