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Theorem fo00 5509
Description: Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
fo00  |-  ( F : (/) -onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )

Proof of Theorem fo00
StepHypRef Expression
1 fofn 5453 . . . . . 6  |-  ( F : (/) -onto-> A  ->  F  Fn  (/) )
2 fn0 5363 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
3 f10 5507 . . . . . . . 8  |-  (/) : (/) -1-1-> A
4 f1eq1 5432 . . . . . . . 8  |-  ( F  =  (/)  ->  ( F : (/) -1-1-> A  <->  (/) : (/) -1-1-> A ) )
53, 4mpbiri 224 . . . . . . 7  |-  ( F  =  (/)  ->  F : (/) -1-1->
A )
62, 5sylbi 187 . . . . . 6  |-  ( F  Fn  (/)  ->  F : (/) -1-1->
A )
71, 6syl 15 . . . . 5  |-  ( F : (/) -onto-> A  ->  F : (/) -1-1->
A )
87ancri 535 . . . 4  |-  ( F : (/) -onto-> A  ->  ( F : (/) -1-1-> A  /\  F : (/)
-onto-> A ) )
9 df-f1o 5262 . . . 4  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F : (/) -1-1-> A  /\  F : (/) -onto-> A ) )
108, 9sylibr 203 . . 3  |-  ( F : (/) -onto-> A  ->  F : (/) -1-1-onto-> A )
11 f1ofo 5479 . . 3  |-  ( F : (/)
-1-1-onto-> A  ->  F : (/) -onto-> A )
1210, 11impbii 180 . 2  |-  ( F : (/) -onto-> A  <->  F : (/) -1-1-onto-> A )
13 f1o00 5508 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
1412, 13bitri 240 1  |-  ( F : (/) -onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623   (/)c0 3455    Fn wfn 5250   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem is referenced by:  fsumf1o  12196  0ramcl  13070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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