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Theorem fn0 5354
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 5333 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 5334 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 4863 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 297 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 411 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 5293 . . . 4 Fun ∅
7 dm0 4859 . . . 4 dom ∅ = ∅
8 df-fn 5238 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 944 . . 3 ∅ Fn ∅
10 fneq1 5323 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 168 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 126 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1364  c0 3437  dom cdm 4644  Rel wrel 4649  Fun wfun 5229   Fn wfn 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-fun 5237  df-fn 5238
This theorem is referenced by:  mpt0  5362  f0  5425  f00  5426  f0bi  5427  f1o00  5515  fo00  5516  tpos0  6300  ixp0x  6753  0fz1  10077
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