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Theorem map0g 8055
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 4066 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6247 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8028 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3syl5ibr 236 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵)))
5 ne0i 4056 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
76exlimdv 2002 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
81, 7syl5bi 232 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴𝑚 𝐵) ≠ ∅))
98necon4d 2948 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6239 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6180 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 248 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8028 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴𝑚 𝐵) ↔ ∅:𝐵𝐴))
1412, 13syl5ibr 236 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴𝑚 𝐵)))
15 ne0i 4056 . . . . 5 (∅ ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴𝑚 𝐵) ≠ ∅))
1716necon2d 2947 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 556 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 6812 . . 3 (𝐴 = ∅ → (𝐴𝑚 𝐵) = (∅ ↑𝑚 𝐵))
20 map0b 8054 . . 3 (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅)
2119, 20sylan9eq 2806 . 2 ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) = ∅)
2218, 21impbid1 215 1 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wex 1845  wcel 2131  wne 2924  c0 4050  {csn 4313   × cxp 5256  wf 6037  (class class class)co 6805  𝑚 cmap 8015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-map 8017
This theorem is referenced by:  map0  8056  mapdom2  8288
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