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Theorem map0g 8442
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 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))

Proof of Theorem map0g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 n0 4309 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6562 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8413 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3syl5ibr 248 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴m 𝐵)))
5 ne0i 4299 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
76exlimdv 1930 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
81, 7syl5bi 244 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴m 𝐵) ≠ ∅))
98necon4d 3040 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6554 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6490 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 260 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8413 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴m 𝐵) ↔ ∅:𝐵𝐴))
1412, 13syl5ibr 248 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴m 𝐵)))
15 ne0i 4299 . . . . 5 (∅ ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴m 𝐵) ≠ ∅))
1716necon2d 3039 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 515 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 7157 . . 3 (𝐴 = ∅ → (𝐴m 𝐵) = (∅ ↑m 𝐵))
20 map0b 8441 . . 3 (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
2119, 20sylan9eq 2876 . 2 ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) = ∅)
2218, 21impbid1 227 1 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  wne 3016  c0 4290  {csn 4560   × cxp 5547  wf 6345  (class class class)co 7150  m cmap 8400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402
This theorem is referenced by:  map0  8445  mapdom2  8682
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