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Theorem f1oun 5355
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 5343 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
2 dff1o4 5343 . . . 4  |-  ( G : C -1-1-onto-> D  <->  ( G  Fn  C  /\  `' G  Fn  D ) )
3 fnun 5199 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  C
)  /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  C )
)
43ex 114 . . . . . 6  |-  ( ( F  Fn  A  /\  G  Fn  C )  ->  ( ( A  i^i  C )  =  (/)  ->  ( F  u.  G )  Fn  ( A  u.  C
) ) )
5 fnun 5199 . . . . . . . 8  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
6 cnvun 4914 . . . . . . . . 9  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
76fneq1i 5187 . . . . . . . 8  |-  ( `' ( F  u.  G
)  Fn  ( B  u.  D )  <->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
85, 7sylibr 133 . . . . . . 7  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  `' ( F  u.  G )  Fn  ( B  u.  D ) )
98ex 114 . . . . . 6  |-  ( ( `' F  Fn  B  /\  `' G  Fn  D
)  ->  ( ( B  i^i  D )  =  (/)  ->  `' ( F  u.  G )  Fn  ( B  u.  D
) ) )
104, 9im2anan9 572 . . . . 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 562 . . . 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 289 . . 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 5343 . . 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 161 . 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 123 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 103    = wceq 1316    u. cun 3039    i^i cin 3040   (/)c0 3333   `'ccnv 4508    Fn wfn 5088   -1-1-onto->wf1o 5092
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-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  f1oprg  5379  unen  6678  zfz1isolem1  10551  ennnfonelemhf1o  11853
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