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Theorem fn0 5909
Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fn0 (𝐹 Fn ∅ ↔ 𝐹 = ∅)

Proof of Theorem fn0
StepHypRef Expression
1 fnrel 5888 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 5889 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 5250 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 500 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 690 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 5853 . . . 4 Fun ∅
7 dm0 5246 . . . 4 dom ∅ = ∅
8 df-fn 5792 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 956 . . 3 ∅ Fn ∅
10 fneq1 5878 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 246 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 197 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  c0 3873  dom cdm 5027  Rel wrel 5032  Fun wfun 5783   Fn wfn 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-fun 5791  df-fn 5792
This theorem is referenced by:  mpt0  5919  f0  5983  f00  5984  f0bi  5985  f1o00  6067  fo00  6068  tpos0  7246  ixp0x  7799  0fz1  12189  hashf1  13052  fuchom  16392  grpinvfvi  17234  mulgfval  17313  mulgfvi  17316  symgplusg  17580  0frgp  17963  invrfval  18444  psrvscafval  19159  tmdgsum  21656  deg1fvi  23593  hon0  27829  fnchoice  37994  dvnprodlem3  38621
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