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Theorem mapdm0 8882
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 5307 . . . . 5 ∅ ∈ V
2 elmapg 8879 . . . . 5 ((𝐵𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
31, 2mpan2 691 . . . 4 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
4 f0bi 6791 . . . 4 (𝑓:∅⟶𝐵𝑓 = ∅)
53, 4bitrdi 287 . . 3 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 = ∅))
6 velsn 4642 . . 3 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
75, 6bitr4di 289 . 2 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 ∈ {∅}))
87eqrdv 2735 1 (𝐵𝑉 → (𝐵m ∅) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  wf 6557  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868
This theorem is referenced by:  map0e  8922  hashmap  14474  ehl0base  25450  1arithidom  33565  repr0  34626  mpct  45206  rrxtopn0  46308  qndenserrnbl  46310  hoicvr  46563  ovn02  46583  ovnhoi  46618  ovnlecvr2  46625  hoiqssbl  46640  hoimbl  46646  0aryfvalel  48555
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