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Theorem mapdm0OLD 39882
Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mapdm0OLD (𝐴𝑉 → (𝐴𝑚 ∅) = {∅})

Proof of Theorem mapdm0OLD
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 0ex 4942 . . . . . 6 ∅ ∈ V
2 elmapg 8036 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
31, 2mpan2 709 . . . . 5 (𝐴𝑉 → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
43biimpa 502 . . . 4 ((𝐴𝑉𝑓 ∈ (𝐴𝑚 ∅)) → 𝑓:∅⟶𝐴)
5 f0bi 6249 . . . 4 (𝑓:∅⟶𝐴𝑓 = ∅)
64, 5sylib 208 . . 3 ((𝐴𝑉𝑓 ∈ (𝐴𝑚 ∅)) → 𝑓 = ∅)
76ralrimiva 3104 . 2 (𝐴𝑉 → ∀𝑓 ∈ (𝐴𝑚 ∅)𝑓 = ∅)
8 f0 6247 . . . . . 6 ∅:∅⟶𝐴
98a1i 11 . . . . 5 (𝐴𝑉 → ∅:∅⟶𝐴)
10 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
111a1i 11 . . . . . 6 (𝐴𝑉 → ∅ ∈ V)
12 elmapg 8036 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (∅ ∈ (𝐴𝑚 ∅) ↔ ∅:∅⟶𝐴))
1310, 11, 12syl2anc 696 . . . . 5 (𝐴𝑉 → (∅ ∈ (𝐴𝑚 ∅) ↔ ∅:∅⟶𝐴))
149, 13mpbird 247 . . . 4 (𝐴𝑉 → ∅ ∈ (𝐴𝑚 ∅))
15 ne0i 4064 . . . 4 (∅ ∈ (𝐴𝑚 ∅) → (𝐴𝑚 ∅) ≠ ∅)
1614, 15syl 17 . . 3 (𝐴𝑉 → (𝐴𝑚 ∅) ≠ ∅)
17 eqsn 4506 . . 3 ((𝐴𝑚 ∅) ≠ ∅ → ((𝐴𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴𝑚 ∅)𝑓 = ∅))
1816, 17syl 17 . 2 (𝐴𝑉 → ((𝐴𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴𝑚 ∅)𝑓 = ∅))
197, 18mpbird 247 1 (𝐴𝑉 → (𝐴𝑚 ∅) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  Vcvv 3340  c0 4058  {csn 4321  wf 6045  (class class class)co 6813  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025
This theorem is referenced by: (None)
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