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Theorem fo00 5711
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 5655 . . . . . 6  |-  ( F : (/) -onto-> A  ->  F  Fn  (/) )
2 fn0 5564 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
3 f10 5709 . . . . . . . 8  |-  (/) : (/) -1-1-> A
4 f1eq1 5634 . . . . . . . 8  |-  ( F  =  (/)  ->  ( F : (/) -1-1-> A  <->  (/) : (/) -1-1-> A ) )
53, 4mpbiri 225 . . . . . . 7  |-  ( F  =  (/)  ->  F : (/) -1-1->
A )
62, 5sylbi 188 . . . . . 6  |-  ( F  Fn  (/)  ->  F : (/) -1-1->
A )
71, 6syl 16 . . . . 5  |-  ( F : (/) -onto-> A  ->  F : (/) -1-1->
A )
87ancri 536 . . . 4  |-  ( F : (/) -onto-> A  ->  ( F : (/) -1-1-> A  /\  F : (/)
-onto-> A ) )
9 df-f1o 5461 . . . 4  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F : (/) -1-1-> A  /\  F : (/) -onto-> A ) )
108, 9sylibr 204 . . 3  |-  ( F : (/) -onto-> A  ->  F : (/) -1-1-onto-> A )
11 f1ofo 5681 . . 3  |-  ( F : (/)
-1-1-onto-> A  ->  F : (/) -onto-> A )
1210, 11impbii 181 . 2  |-  ( F : (/) -onto-> A  <->  F : (/) -1-1-onto-> A )
13 f1o00 5710 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
1412, 13bitri 241 1  |-  ( F : (/) -onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652   (/)c0 3628    Fn wfn 5449   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453
This theorem is referenced by:  fsumf1o  12517  0ramcl  13391  fprodf1o  25272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461
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