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Theorem map0g 7783
 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 3890 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6007 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 7757 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3syl5ibr 235 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵)))
5 ne0i 3880 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
64, 5syl6 34 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
76exlimdv 1848 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
81, 7syl5bi 231 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴𝑚 𝐵) ≠ ∅))
98necon4d 2806 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐴 = ∅))
10 f0 5999 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 5940 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 247 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 7757 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴𝑚 𝐵) ↔ ∅:𝐵𝐴))
1412, 13syl5ibr 235 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴𝑚 𝐵)))
15 ne0i 3880 . . . . 5 (∅ ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
1614, 15syl6 34 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴𝑚 𝐵) ≠ ∅))
1716necon2d 2805 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 554 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 6556 . . 3 (𝐴 = ∅ → (𝐴𝑚 𝐵) = (∅ ↑𝑚 𝐵))
20 map0b 7782 . . 3 (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅)
2119, 20sylan9eq 2664 . 2 ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) = ∅)
2218, 21impbid1 214 1 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {csn 4125   × cxp 5036  ⟶wf 5800  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  map0  7784  mapdom2  8016
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