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Theorem dff1o2 5827
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )

Proof of Theorem dff1o2
StepHypRef Expression
1 df-f1o 5601 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 df-f1 5599 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
3 df-fo 5600 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 697 . . 3  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( ( F : A --> B  /\  Fun  `' F )  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
5 anass 649 . . . 4  |-  ( ( ( F : A --> B  /\  Fun  `' F
)  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) ) )
6 3anan12 986 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
76anbi1i 695 . . . . 5  |-  ( ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  /\  F : A --> B )  <->  ( ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) )  /\  F : A
--> B ) )
8 eqimss 3551 . . . . . . . 8  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
9 df-f 5598 . . . . . . . . 9  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
109biimpri 206 . . . . . . . 8  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  F : A --> B )
118, 10sylan2 474 . . . . . . 7  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
12113adant2 1015 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  ->  F : A --> B )
1312pm4.71i 632 . . . . 5  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  /\  F : A --> B ) )
14 ancom 450 . . . . 5  |-  ( ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )  <->  ( ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) )  /\  F : A
--> B ) )
157, 13, 143bitr4ri 278 . . . 4  |-  ( ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
165, 15bitri 249 . . 3  |-  ( ( ( F : A --> B  /\  Fun  `' F
)  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
174, 16bitri 249 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
181, 17bitri 249 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    C_ wss 3471   `'ccnv 5007   ran crn 5009   Fun wfun 5588    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   -onto->wfo 5592   -1-1-onto->wf1o 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601
This theorem is referenced by:  dff1o3  5828  dff1o4  5830  f1orn  5832  tz7.49c  7129  fiint  7815  symgfixelsi  16587  dfrelog  23079  adj1o  26940  f1mptrn  27620  esumc  28225  stoweidlem39  32024
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