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Theorem f1oun 5462
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 5450 . . . 4  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
2 dff1o4 5450 . . . 4  |-  ( G : C -1-1-onto-> D  <->  ( G  Fn  C  /\  `' G  Fn  D ) )
3 fnun 5304 . . . . . . 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 5304 . . . . . . . 8  |-  ( ( ( `' F  Fn  B  /\  `' G  Fn  D )  /\  ( B  i^i  D )  =  (/) )  ->  ( `' F  u.  `' G
)  Fn  ( B  u.  D ) )
6 cnvun 5016 . . . . . . . . 9  |-  `' ( F  u.  G )  =  ( `' F  u.  `' G )
76fneq1i 5292 . . . . . . . 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 593 . . . . 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 583 . . . 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 5450 . . 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 1348    u. cun 3119    i^i cin 3120   (/)c0 3414   `'ccnv 4610    Fn wfn 5193   -1-1-onto->wf1o 5197
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205
This theorem is referenced by:  f1oprg  5486  unen  6794  zfz1isolem1  10775  ennnfonelemhf1o  12368
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