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Theorem map0g 8923
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 4359 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6798 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8878 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3imbitrrid 246 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴m 𝐵)))
5 ne0i 4347 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
76exlimdv 1931 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
81, 7biimtrid 242 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴m 𝐵) ≠ ∅))
98necon4d 2962 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6790 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6718 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 258 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8878 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴m 𝐵) ↔ ∅:𝐵𝐴))
1412, 13imbitrrid 246 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴m 𝐵)))
15 ne0i 4347 . . . . 5 (∅ ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴m 𝐵) ≠ ∅))
1716necon2d 2961 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 512 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 7438 . . 3 (𝐴 = ∅ → (𝐴m 𝐵) = (∅ ↑m 𝐵))
20 map0b 8922 . . 3 (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
2119, 20sylan9eq 2795 . 2 ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) = ∅)
2218, 21impbid1 225 1 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  c0 4339  {csn 4631   × cxp 5687  wf 6559  (class class class)co 7431  m cmap 8865
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  map0  8926  mapdom2  9187  map0cor  48685
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