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Theorem fo00 5525
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 5469 . . . . . 6  |-  ( F : (/) -onto-> A  ->  F  Fn  (/) )
2 fn0 5379 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
3 f10 5523 . . . . . . . 8  |-  (/) : (/) -1-1-> A
4 f1eq1 5448 . . . . . . . 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 5278 . . . 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 5495 . . 3  |-  ( F : (/)
-1-1-onto-> A  ->  F : (/) -onto-> A )
1210, 11impbii 180 . 2  |-  ( F : (/) -onto-> A  <->  F : (/) -1-1-onto-> A )
13 f1o00 5524 . 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 1632   (/)c0 3468    Fn wfn 5266   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270
This theorem is referenced by:  fsumf1o  12212  0ramcl  13086  fprodf1o  24172
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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