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Theorem fn0 5119
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 5098 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 5099 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 4642 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 291 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 403 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 5058 . . . 4 Fun ∅
7 dm0 4638 . . . 4 dom ∅ = ∅
8 df-fn 5005 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 888 . . 3 ∅ Fn ∅
10 fneq1 5088 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 166 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 124 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1289  c0 3284  dom cdm 4428  Rel wrel 4433  Fun wfun 4996   Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-fun 5004  df-fn 5005
This theorem is referenced by:  mpt0  5127  f0  5185  f00  5186  f0bi  5187  f1o00  5272  fo00  5273  tpos0  6021  0fz1  9428
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