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Theorem fo00 6172
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 6117 . . . . . 6 |- ( F : (/) -onto-> A -> F Fn (/) )
2 fn0 6011 . . . . . . 7 |- ( F Fn (/) <-> F = (/) )
3 f10 6169 . . . . . . . 8 |- (/) : (/) -1-1-> A
4 f1eq1 6096 . . . . . . . 8 |- ( F = (/) -> ( F : (/) -1-1-> A <-> (/) : (/) -1-1-> A ) )
53, 4mpbiri 248 . . . . . . 7 |- ( F = (/) -> F : (/) -1-1-> A )
62, 5sylbi 207 . . . . . 6 |- ( F Fn (/) -> F : (/) -1-1-> A )
71, 6syl 17 . . . . 5 |- ( F : (/) -onto-> A -> F : (/) -1-1-> A )
87ancri 575 . . . 4 |- ( F : (/) -onto-> A -> ( F : (/) -1-1-> A /\ F : (/) -onto-> A ) )
9 df-f1o 5895 . . . 4 |- ( F : (/) -1-1-onto-> A <-> ( F : (/) -1-1-> A /\ F : (/) -onto-> A ) )
108, 9sylibr 224 . . 3 |- ( F : (/) -onto-> A -> F : (/) -1-1-onto-> A )
11 f1ofo 6144 . . 3 |- ( F : (/) -1-1-onto-> A -> F : (/) -onto-> A )
1210, 11impbii 199 . 2 |- ( F : (/) -onto-> A <-> F : (/) -1-1-onto-> A )
13 f1o00 6171 . 2 |- ( F : (/) -1-1-onto-> A <-> ( F = (/) /\ A = (/) ) )
1412, 13bitri 264 1 |- ( F : (/) -onto-> A <-> ( F = (/) /\ A = (/) ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   (/)c0 3915   Fn wfn 5883   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895
This theorem is referenced by:  fsumf1o  14454  fprodf1o  14676  0ramcl  15727
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