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Theorem mapdm0 8404
 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 5192 . . . . 5 ∅ ∈ V
2 elmapg 8402 . . . . 5 ((𝐵𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
31, 2mpan2 690 . . . 4 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
4 f0bi 6543 . . . 4 (𝑓:∅⟶𝐵𝑓 = ∅)
53, 4syl6bb 290 . . 3 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 = ∅))
6 velsn 4564 . . 3 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
75, 6syl6bbr 292 . 2 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 ∈ {∅}))
87eqrdv 2822 1 (𝐵𝑉 → (𝐵m ∅) = {∅})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  Vcvv 3479  ∅c0 4274  {csn 4548  ⟶wf 6332  (class class class)co 7138   ↑m cmap 8389 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391 This theorem is referenced by:  map0e  8429  hashmap  13790  ehl0base  24009  repr0  31900  mpct  41671  rrxtopn0  42777  qndenserrnbl  42779  hoicvr  43029  ovn02  43049  ovnhoi  43084  ovnlecvr2  43091  hoiqssbl  43106  hoimbl  43112  0aryfvalel  44888
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