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Theorem f10 6640
Description: The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
Assertion
Ref Expression
f10 ∅:∅–1-1𝐴

Proof of Theorem f10
StepHypRef Expression
1 f0 6553 . 2 ∅:∅⟶𝐴
2 funcnv0 6413 . 2 Fun
3 df-f1 6353 . 2 (∅:∅–1-1𝐴 ↔ (∅:∅⟶𝐴 ∧ Fun ∅))
41, 2, 3mpbir2an 707 1 ∅:∅–1-1𝐴
Colors of variables: wff setvar class
Syntax hints:  c0 4288  ccnv 5547  Fun wfun 6342  wf 6344  1-1wf1 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353
This theorem is referenced by:  f10d  6641  fo00  6643  marypha1lem  8885  hashf1  13803  usgr0  26952
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