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Theorem fn0 6172
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 6150 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 6151 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 5498 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 503 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 696 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 6115 . . . 4 Fun ∅
7 dm0 5494 . . . 4 dom ∅ = ∅
8 df-fn 6052 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 993 . . 3 ∅ Fn ∅
10 fneq1 6140 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 248 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 199 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  c0 4058  dom cdm 5266  Rel wrel 5271  Fun wfun 6043   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-fun 6051  df-fn 6052
This theorem is referenced by:  mpt0  6182  f0  6247  f00  6248  f0bi  6249  f1o00  6333  fo00  6334  tpos0  7552  ixp0x  8104  0fz1  12574  hashf1  13453  fuchom  16842  grpinvfvi  17684  mulgfval  17763  mulgfvi  17766  symgplusg  18029  0frgp  18412  invrfval  18893  psrvscafval  19612  tmdgsum  22120  deg1fvi  24064  hon0  28982  fnchoice  39705  dvnprodlem3  40684
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