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Theorem fn0 5212
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 5191 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 5192 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 4727 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 295 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 408 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 5151 . . . 4 Fun ∅
7 dm0 4723 . . . 4 dom ∅ = ∅
8 df-fn 5096 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 911 . . 3 ∅ Fn ∅
10 fneq1 5181 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 167 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 125 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1316  c0 3333  dom cdm 4509  Rel wrel 4514  Fun wfun 5087   Fn wfn 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  mpt0  5220  f0  5283  f00  5284  f0bi  5285  f1o00  5370  fo00  5371  tpos0  6139  ixp0x  6588  0fz1  9793
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