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Theorem fn0 5336
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 5315 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 5316 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 4846 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 297 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 411 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 5275 . . . 4 Fun ∅
7 dm0 4842 . . . 4 dom ∅ = ∅
8 df-fn 5220 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 942 . . 3 ∅ Fn ∅
10 fneq1 5305 . . 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 1353  c0 3423  dom cdm 4627  Rel wrel 4632  Fun wfun 5211   Fn wfn 5212
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-fun 5219  df-fn 5220
This theorem is referenced by:  mpt0  5344  f0  5407  f00  5408  f0bi  5409  f1o00  5497  fo00  5498  tpos0  6275  ixp0x  6726  0fz1  10045
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