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Theorem mapdm0 8861
Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)
Assertion
Ref Expression
mapdm0 (𝐵𝑉 → (𝐵m ∅) = {∅})

Proof of Theorem mapdm0
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 0ex 5308 . . . . 5 ∅ ∈ V
2 elmapg 8858 . . . . 5 ((𝐵𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
31, 2mpan2 689 . . . 4 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
4 f0bi 6780 . . . 4 (𝑓:∅⟶𝐵𝑓 = ∅)
53, 4bitrdi 286 . . 3 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 = ∅))
6 velsn 4646 . . 3 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
75, 6bitr4di 288 . 2 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 ∈ {∅}))
87eqrdv 2723 1 (𝐵𝑉 → (𝐵m ∅) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322  {csn 4630  wf 6545  (class class class)co 7419  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847
This theorem is referenced by:  map0e  8901  hashmap  14435  ehl0base  25393  1arithidom  33354  repr0  34376  mpct  44715  rrxtopn0  45821  qndenserrnbl  45823  hoicvr  46076  ovn02  46096  ovnhoi  46131  ovnlecvr2  46138  hoiqssbl  46153  hoimbl  46159  0aryfvalel  47895
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