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Theorem mapdm0 8783
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 5265 . . . . 5 ∅ ∈ V
2 elmapg 8781 . . . . 5 ((𝐵𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
31, 2mpan2 690 . . . 4 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓:∅⟶𝐵))
4 f0bi 6726 . . . 4 (𝑓:∅⟶𝐵𝑓 = ∅)
53, 4bitrdi 287 . . 3 (𝐵𝑉 → (𝑓 ∈ (𝐵m ∅) ↔ 𝑓 = ∅))
6 velsn 4603 . . 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 3444  c0 4283  {csn 4587  wf 6493  (class class class)co 7358  m cmap 8768
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  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-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770
This theorem is referenced by:  map0e  8823  hashmap  14341  ehl0base  24796  repr0  33281  mpct  43509  rrxtopn0  44620  qndenserrnbl  44622  hoicvr  44875  ovn02  44895  ovnhoi  44930  ovnlecvr2  44937  hoiqssbl  44952  hoimbl  44958  0aryfvalel  46806
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