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	⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅})

Proof of Theorem mapdm0OLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4942	. . . . . 6 ⊢ ∅ ∈ V
2		elmapg 8036	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
3	1, 2	mpan2 709	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
4	3	biimpa 502	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓:∅⟶𝐴)
5		f0bi 6249	. . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅)
6	4, 5	sylib 208	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑓 ∈ (𝐴 ↑𝑚 ∅)) → 𝑓 = ∅)
7	6	ralrimiva 3104	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑓 ∈ (𝐴 ↑𝑚 ∅)𝑓 = ∅)
8		f0 6247	. . . . . 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 8036	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴))
13	10, 11, 12	syl2anc 696	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∅ ∈ (𝐴 ↑𝑚 ∅) ↔ ∅:∅⟶𝐴))
14	9, 13	mpbird 247	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ (𝐴 ↑𝑚 ∅))
15		ne0i 4064	. . . 4 ⊢ (∅ ∈ (𝐴 ↑𝑚 ∅) → (𝐴 ↑𝑚 ∅) ≠ ∅)
16	14, 15	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) ≠ ∅)
17		eqsn 4506	. . 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 247	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = {∅})

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)