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Theorem mapdm0OLD 40137
 Description: Obsolete version of mapdm0 8110 as of 3-Dec-2021. (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 4984 . . . . . 6 ∅ ∈ V
2 elmapg 8108 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
31, 2mpan2 683 . . . . 5 (𝐴𝑉 → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
43biimpa 469 . . . 4 ((𝐴𝑉𝑓 ∈ (𝐴𝑚 ∅)) → 𝑓:∅⟶𝐴)
5 f0bi 6303 . . . 4 (𝑓:∅⟶𝐴𝑓 = ∅)
64, 5sylib 210 . . 3 ((𝐴𝑉𝑓 ∈ (𝐴𝑚 ∅)) → 𝑓 = ∅)
76ralrimiva 3147 . 2 (𝐴𝑉 → ∀𝑓 ∈ (𝐴𝑚 ∅)𝑓 = ∅)
8 f0 6301 . . . . . 6 ∅:∅⟶𝐴
98a1i 11 . . . . 5 (𝐴𝑉 → ∅:∅⟶𝐴)
10 id 22 . . . . . 6 (𝐴𝑉𝐴𝑉)
111a1i 11 . . . . . 6 (𝐴𝑉 → ∅ ∈ V)
12 elmapg 8108 . . . . . 6 ((𝐴𝑉 ∧ ∅ ∈ V) → (∅ ∈ (𝐴𝑚 ∅) ↔ ∅:∅⟶𝐴))
1310, 11, 12syl2anc 580 . . . . 5 (𝐴𝑉 → (∅ ∈ (𝐴𝑚 ∅) ↔ ∅:∅⟶𝐴))
149, 13mpbird 249 . . . 4 (𝐴𝑉 → ∅ ∈ (𝐴𝑚 ∅))
15 ne0i 4121 . . . 4 (∅ ∈ (𝐴𝑚 ∅) → (𝐴𝑚 ∅) ≠ ∅)
1614, 15syl 17 . . 3 (𝐴𝑉 → (𝐴𝑚 ∅) ≠ ∅)
17 eqsn 4548 . . 3 ((𝐴𝑚 ∅) ≠ ∅ → ((𝐴𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴𝑚 ∅)𝑓 = ∅))
1816, 17syl 17 . 2 (𝐴𝑉 → ((𝐴𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴𝑚 ∅)𝑓 = ∅))
197, 18mpbird 249 1 (𝐴𝑉 → (𝐴𝑚 ∅) = {∅})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ≠ wne 2971  ∀wral 3089  Vcvv 3385  ∅c0 4115  {csn 4368  ⟶wf 6097  (class class class)co 6878   ↑𝑚 cmap 8095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097 This theorem is referenced by: (None)
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