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Theorem f0 6554
Description: The empty function. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
f0 ∅:∅⟶𝐴

Proof of Theorem f0
StepHypRef Expression
1 eqid 2821 . . 3 ∅ = ∅
2 fn0 6473 . . 3 (∅ Fn ∅ ↔ ∅ = ∅)
31, 2mpbir 232 . 2 ∅ Fn ∅
4 rn0 5790 . . 3 ran ∅ = ∅
5 0ss 4349 . . 3 ∅ ⊆ 𝐴
64, 5eqsstri 4000 . 2 ran ∅ ⊆ 𝐴
7 df-f 6353 . 2 (∅:∅⟶𝐴 ↔ (∅ Fn ∅ ∧ ran ∅ ⊆ 𝐴))
83, 6, 7mpbir2an 707 1 ∅:∅⟶𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wss 3935  c0 4290  ran crn 5550   Fn wfn 6344  wf 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 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-fun 6351  df-fn 6352  df-f 6353
This theorem is referenced by:  f00  6555  f0bi  6556  f10  6641  map0g  8438  ac6sfi  8751  oif  8983  wrd0  13879  0csh0  14145  ram0  16348  0ssc  17097  0subcat  17098  gsum0  17884  ga0  18368  0frgp  18836  ptcmpfi  22351  0met  22905  perfdvf  24430  uhgr0e  26784  uhgr0  26786  griedg0prc  26974  locfinref  31005  matunitlindf  34772  poimirlem28  34802  climlimsupcex  41930  0cnf  42040  dvnprodlem3  42113  sge00  42539  hoidmvlelem3  42760
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