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	⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅})

Proof of Theorem mapdm0OLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4984	. . . . . 6 ⊢ ∅ ∈ V
2		elmapg 8108	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
3	1, 2	mpan2 683	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
4	3	biimpa 469	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓:∅⟶𝐴)
5		f0bi 6303	. . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅)
6	4, 5	sylib 210	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓 = ∅)
7	6	ralrimiva 3147	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅)
8		f0 6301	. . . . . 6 ⊢ ∅:∅⟶𝐴
9	8	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∅:∅⟶𝐴)
10		id 22	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
11	1	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
12		elmapg 8108	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴))
13	10, 11, 12	syl2anc 580	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴))
14	9, 13	mpbird 249	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (𝐴 ↑𝑚 ∅))
15		ne0i 4121	. . . 4 ⊢ (∅ ∈ (𝐴 ↑𝑚 ∅) → (𝐴 ↑𝑚 ∅) ≠ ∅)
16	14, 15	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) ≠ ∅)
17		eqsn 4548	. . 3 ⊢ ((𝐴 ↑𝑚 ∅) ≠ ∅ → ((𝐴 ↑𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅))
18	16, 17	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ↑𝑚 ∅) = {∅} ↔ ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅))
19	7, 18	mpbird 249	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅})

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)