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Theorem f10 5590
Description: The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
Assertion
Ref Expression
f10  |-  (/) : (/) -1-1-> A

Proof of Theorem f10
StepHypRef Expression
1 f0 5508 . 2  |-  (/) : (/) --> A
2 fun0 5389 . . 3  |-  Fun  (/)
3 cnv0 5166 . . . 4  |-  `' (/)  =  (/)
43funeqi 5357 . . 3  |-  ( Fun  `' (/)  <->  Fun  (/) )
52, 4mpbir 200 . 2  |-  Fun  `' (/)
6 df-f1 5342 . 2  |-  ( (/) :
(/) -1-1-> A  <->  ( (/) : (/) --> A  /\  Fun  `' (/) ) )
71, 5, 6mpbir2an 886 1  |-  (/) : (/) -1-1-> A
Colors of variables: wff set class
Syntax hints:   (/)c0 3531   `'ccnv 4770   Fun wfun 5331   -->wf 5333   -1-1->wf1 5334
This theorem is referenced by:  fo00  5592  marypha1lem  7276  hashf1  11491  usgra0  27545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342
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