Theorem map0g 8870

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 4308	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝐴)
2		fconst6g 6757	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}):𝐵⟶𝐴)
3		elmapg 8824	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵) ↔ (𝐵 × {𝑓}):𝐵⟶𝐴))
4	2, 3	imbitrrid 249	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵)))
5		ne0i 4296	. . . . . . 7 ⊢ ((𝐵 × {𝑓}) ∈ (𝐴 ↑m 𝐵) → (𝐴 ↑m 𝐵) ≠ ∅)
6	4, 5	syl6 36	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑓 ∈ 𝐴 → (𝐴 ↑m 𝐵) ≠ ∅))
7	6	exlimdv 1956	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑓 𝑓 ∈ 𝐴 → (𝐴 ↑m 𝐵) ≠ ∅))
8	1, 7	biimtrid 245	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ ∅ → (𝐴 ↑m 𝐵) ≠ ∅))
9	8	necon4d 2984	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → 𝐴 = ∅))
10		f0 6749	. . . . . . 7 ⊢ ∅:∅⟶𝐴
11		feq2 6674	. . . . . . 7 ⊢ (𝐵 = ∅ → (∅:𝐵⟶𝐴 ↔ ∅:∅⟶𝐴))
12	10, 11	mpbiri 261	. . . . . 6 ⊢ (𝐵 = ∅ → ∅:𝐵⟶𝐴)
13		elmapg 8824	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∅ ∈ (𝐴 ↑m 𝐵) ↔ ∅:𝐵⟶𝐴))
14	12, 13	imbitrrid 249	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴 ↑m 𝐵)))
15		ne0i 4296	. . . . 5 ⊢ (∅ ∈ (𝐴 ↑m 𝐵) → (𝐴 ↑m 𝐵) ≠ ∅)
16	14, 15	syl6 36	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 = ∅ → (𝐴 ↑m 𝐵) ≠ ∅))
17	16	necon2d 2983	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → 𝐵 ≠ ∅))
18	9, 17	jcad 521	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19		oveq1 7407	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐵) = (∅ ↑m 𝐵))
20		map0b 8869	. . 3 ⊢ (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
21	19, 20	sylan9eq 2820	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) = ∅)
22	18, 21	impbid1 228	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∅c0 4288  {csn 4585   × cxp 5650  ⟶wf 6521  (class class class)co 7400   ↑_m cmap 8812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814

This theorem is referenced by:  map0  8873  mapdom2  9124  map0cor  49484