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Theorem mapdm0 8023
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 4921 . . . . 5 ∅ ∈ V
2 elmapg 8021 . . . . 5 ((𝐵𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐵𝑚 ∅) ↔ 𝑓:∅⟶𝐵))
31, 2mpan2 663 . . . 4 (𝐵𝑉 → (𝑓 ∈ (𝐵𝑚 ∅) ↔ 𝑓:∅⟶𝐵))
4 f0bi 6228 . . . 4 (𝑓:∅⟶𝐵𝑓 = ∅)
53, 4syl6bb 276 . . 3 (𝐵𝑉 → (𝑓 ∈ (𝐵𝑚 ∅) ↔ 𝑓 = ∅))
6 vex 3352 . . . 4 𝑓 ∈ V
76elsn 4329 . . 3 (𝑓 ∈ {∅} ↔ 𝑓 = ∅)
85, 7syl6bbr 278 . 2 (𝐵𝑉 → (𝑓 ∈ (𝐵𝑚 ∅) ↔ 𝑓 ∈ {∅}))
98eqrdv 2768 1 (𝐵𝑉 → (𝐵𝑚 ∅) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  {csn 4314  wf 6027  (class class class)co 6792  𝑚 cmap 8008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010
This theorem is referenced by:  map0e  8046  hashmap  13423  repr0  31023  mpct  39905  rrxtopn0  41024  qndenserrnbl  41026  hoicvr  41276  ovn02  41296  ovnhoi  41331  ovnlecvr2  41338  hoiqssbl  41353  hoimbl  41359
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