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Theorem f10 6081
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 5999 . 2 ∅:∅⟶𝐴
2 fun0 5868 . . 3 Fun ∅
3 cnv0 5454 . . . 4 ∅ = ∅
43funeqi 5824 . . 3 (Fun ∅ ↔ Fun ∅)
52, 4mpbir 220 . 2 Fun
6 df-f1 5809 . 2 (∅:∅–1-1𝐴 ↔ (∅:∅⟶𝐴 ∧ Fun ∅))
71, 5, 6mpbir2an 957 1 ∅:∅–1-1𝐴
Colors of variables: wff setvar class
Syntax hints:  c0 3874  ccnv 5037  Fun wfun 5798  wf 5800  1-1wf1 5801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809
This theorem is referenced by:  f10d  6082  fo00  6084  marypha1lem  8222  hashf1  13098  usgr0  40469
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