Theorem map0g 8924

Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
map0g	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Proof of Theorem map0g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4353	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴)
2		fconst6g 6797	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}):𝐵⟶𝐴)
3		elmapg 8879	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵) ↔ (𝐵 × {𝑓}):𝐵⟶𝐴))
4	2, 3	imbitrrid 246	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵)))
5		ne0i 4341	. . . . . . 7 ⊢ ((𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵) → (𝐴 ↑m 𝐵) ≠ ∅)
6	4, 5	syl6 35	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐴 ↑m 𝐵) ≠ ∅))
7	6	exlimdv 1933	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓 ∈ 𝐴 → (𝐴 ↑m 𝐵) ≠ ∅))
8	1, 7	biimtrid 242	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ ∅ → (𝐴 ↑m 𝐵) ≠ ∅))
9	8	necon4d 2964	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → 𝐴 = ∅))
10		f0 6789	. . . . . . 7 ⊢ ∅:∅⟶𝐴
11		feq2 6717	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅:𝐵⟶𝐴 ↔ ∅:∅⟶𝐴))
12	10, 11	mpbiri 258	. . . . . 6 ⊢ (𝐵 = ∅ → ∅:𝐵⟶𝐴)
13		elmapg 8879	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∅ ∈ (𝐴 ↑m 𝐵) ↔ ∅:𝐵⟶𝐴))
14	12, 13	imbitrrid 246	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴 ↑m 𝐵)))
15		ne0i 4341	. . . . 5 ⊢ (∅ ∈ (𝐴 ↑m 𝐵) → (𝐴 ↑m 𝐵) ≠ ∅)
16	14, 15	syl6 35	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → (𝐴 ↑m 𝐵) ≠ ∅))
17	16	necon2d 2963	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → 𝐵 ≠ ∅))
18	9, 17	jcad 512	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19		oveq1 7438	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐵) = (∅ ↑m 𝐵))
20		map0b 8923	. . 3 ⊢ (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
21	19, 20	sylan9eq 2797	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) = ∅)
22	18, 21	impbid1 225	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333  {csn 4626   × cxp 5683  ⟶wf 6557  (class class class)co 7431   ↑_m cmap 8866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868

This theorem is referenced by:  map0  8927  mapdom2  9188  map0cor  48764