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Theorem fn0 4961
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 4940 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 4941 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 4496 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 281 . . 3 ((Rel 𝐹 dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 391 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 4900 . . . 4 Fun ∅
7 dm0 4492 . . . 4 dom ∅ = ∅
8 df-fn 4848 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ dom ∅ = ∅))
96, 7, 8mpbir2an 848 . . 3 ∅ Fn ∅
10 fneq1 4930 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 157 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 117 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242  c0 3218  dom cdm 4288  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848
This theorem is referenced by:  mpt0  4969  f0  5023  f00  5024  f1o00  5104  fo00  5105  tpos0  5830  0fz1  8679
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