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Theorem fn0 6633
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 6605 . . 3 (𝐹 Fn ∅ → Rel 𝐹)
2 fndm 6606 . . 3 (𝐹 Fn ∅ → dom 𝐹 = ∅)
3 reldm0 5884 . . . 4 (Rel 𝐹 → (𝐹 = ∅ ↔ dom 𝐹 = ∅))
43biimpar 479 . . 3 ((Rel 𝐹 ∧ dom 𝐹 = ∅) → 𝐹 = ∅)
51, 2, 4syl2anc 585 . 2 (𝐹 Fn ∅ → 𝐹 = ∅)
6 fun0 6567 . . . 4 Fun ∅
7 dm0 5877 . . . 4 dom ∅ = ∅
8 df-fn 6500 . . . 4 (∅ Fn ∅ ↔ (Fun ∅ ∧ dom ∅ = ∅))
96, 7, 8mpbir2an 710 . . 3 ∅ Fn ∅
10 fneq1 6594 . . 3 (𝐹 = ∅ → (𝐹 Fn ∅ ↔ ∅ Fn ∅))
119, 10mpbiri 258 . 2 (𝐹 = ∅ → 𝐹 Fn ∅)
125, 11impbii 208 1 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  c0 4283  dom cdm 5634  Rel wrel 5639  Fun wfun 6491   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500
This theorem is referenced by:  mpt0  6644  f0  6724  f00  6725  f0bi  6726  f1o00  6820  fo00  6821  tpos0  8188  ixp0x  8867  0fz1  13467  hashf1  14362  fuchom  17854  fuchomOLD  17855  grpinvfvi  18798  mulgfval  18879  mulgfvalALT  18880  mulgfvi  18883  0frgp  19566  invrfval  20107  psrvscafval  21374  tmdgsum  23462  deg1fvi  25466  hon0  30777  fnchoice  43322  dvnprodlem3  44275
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