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Theorem mapdm0 8836
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 8833 . . . . 5 ((𝐵𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
31, 2mpan2 690 . . . 4 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
4 f0bi 6775 . . . 4 (𝑓:∅⟶𝐵𝑓 = ∅)
53, 4bitrdi 287 . . 3 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 = ∅))
6 velsn 4645 . . 3 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
75, 6bitr4di 289 . 2 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 ∈ {∅}))
87eqrdv 2731 1 (𝐵𝑉 → (𝐵m ∅) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323  {csn 4629  wf 6540  (class class class)co 7409  m cmap 8820
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822
This theorem is referenced by:  map0e  8876  hashmap  14395  ehl0base  24933  repr0  33654  mpct  43948  rrxtopn0  45057  qndenserrnbl  45059  hoicvr  45312  ovn02  45332  ovnhoi  45367  ovnlecvr2  45374  hoiqssbl  45389  hoimbl  45395  0aryfvalel  47368
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