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Theorem fo00 5785
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 5733 . . . . . 6  |-  ( F : (/) -onto-> A  ->  F  Fn  (/) )
2 fn0 5641 . . . . . . 7  |-  ( F  Fn  (/)  <->  F  =  (/) )
3 f10 5783 . . . . . . . 8  |-  (/) : (/) -1-1-> A
4 f1eq1 5712 . . . . . . . 8  |-  ( F  =  (/)  ->  ( F : (/) -1-1-> A  <->  (/) : (/) -1-1-> A ) )
53, 4mpbiri 233 . . . . . . 7  |-  ( F  =  (/)  ->  F : (/) -1-1->
A )
62, 5sylbi 195 . . . . . 6  |-  ( F  Fn  (/)  ->  F : (/) -1-1->
A )
71, 6syl 16 . . . . 5  |-  ( F : (/) -onto-> A  ->  F : (/) -1-1->
A )
87ancri 552 . . . 4  |-  ( F : (/) -onto-> A  ->  ( F : (/) -1-1-> A  /\  F : (/)
-onto-> A ) )
9 df-f1o 5536 . . . 4  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F : (/) -1-1-> A  /\  F : (/) -onto-> A ) )
108, 9sylibr 212 . . 3  |-  ( F : (/) -onto-> A  ->  F : (/) -1-1-onto-> A )
11 f1ofo 5759 . . 3  |-  ( F : (/)
-1-1-onto-> A  ->  F : (/) -onto-> A )
1210, 11impbii 188 . 2  |-  ( F : (/) -onto-> A  <->  F : (/) -1-1-onto-> A )
13 f1o00 5784 . 2  |-  ( F : (/)
-1-1-onto-> A 
<->  ( F  =  (/)  /\  A  =  (/) ) )
1412, 13bitri 249 1  |-  ( F : (/) -onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   (/)c0 3748    Fn wfn 5524   -1-1->wf1 5526   -onto->wfo 5527   -1-1-onto->wf1o 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536
This theorem is referenced by:  fsumf1o  13321  0ramcl  14205  fprodf1o  27623
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