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

Proof of Theorem map0g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 n0 4353 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6797 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8879 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3imbitrrid 246 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴m 𝐵)))
5 ne0i 4341 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
76exlimdv 1933 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
81, 7biimtrid 242 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴m 𝐵) ≠ ∅))
98necon4d 2964 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6789 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6717 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 258 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8879 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴m 𝐵) ↔ ∅:𝐵𝐴))
1412, 13imbitrrid 246 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴m 𝐵)))
15 ne0i 4341 . . . . 5 (∅ ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴m 𝐵) ≠ ∅))
1716necon2d 2963 . . 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 8923 . . 3 (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
2119, 20sylan9eq 2797 . 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 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
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