Theorem map0g 8894

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 4342	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴)
2		fconst6g 6780	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}):𝐵⟶𝐴)
3		elmapg 8849	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵) ↔ (𝐵 × {𝑓}):𝐵⟶𝐴))
4	2, 3	imbitrrid 245	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵)))
5		ne0i 4330	. . . . . . 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 1929	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓 ∈ 𝐴 → (𝐴 ↑m 𝐵) ≠ ∅))
8	1, 7	biimtrid 241	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ ∅ → (𝐴 ↑m 𝐵) ≠ ∅))
9	8	necon4d 2959	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → 𝐴 = ∅))
10		f0 6772	. . . . . . 7 ⊢ ∅:∅⟶𝐴
11		feq2 6698	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅:𝐵⟶𝐴 ↔ ∅:∅⟶𝐴))
12	10, 11	mpbiri 258	. . . . . 6 ⊢ (𝐵 = ∅ → ∅:𝐵⟶𝐴)
13		elmapg 8849	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∅ ∈ (𝐴 ↑m 𝐵) ↔ ∅:𝐵⟶𝐴))
14	12, 13	imbitrrid 245	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴 ↑m 𝐵)))
15		ne0i 4330	. . . . 5 ⊢ (∅ ∈ (𝐴 ↑m 𝐵) → (𝐴 ↑m 𝐵) ≠ ∅)
16	14, 15	syl6 35	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → (𝐴 ↑m 𝐵) ≠ ∅))
17	16	necon2d 2958	. . 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 7421	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐵) = (∅ ↑m 𝐵))
20		map0b 8893	. . 3 ⊢ (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
21	19, 20	sylan9eq 2787	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) = ∅)
22	18, 21	impbid1 224	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2935  ∅c0 4318  {csn 4624   × cxp 5670  ⟶wf 6538  (class class class)co 7414   ↑_m cmap 8836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8838

This theorem is referenced by:  map0  8897  mapdom2  9164  map0cor  47830