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Theorem map0g 8874
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 4338 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6770 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8829 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3imbitrrid 245 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴m 𝐵)))
5 ne0i 4326 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
76exlimdv 1928 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
81, 7biimtrid 241 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴m 𝐵) ≠ ∅))
98necon4d 2956 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6762 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6689 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 258 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8829 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴m 𝐵) ↔ ∅:𝐵𝐴))
1412, 13imbitrrid 245 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴m 𝐵)))
15 ne0i 4326 . . . . 5 (∅ ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴m 𝐵) ≠ ∅))
1716necon2d 2955 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 512 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 7408 . . 3 (𝐴 = ∅ → (𝐴m 𝐵) = (∅ ↑m 𝐵))
20 map0b 8873 . . 3 (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
2119, 20sylan9eq 2784 . 2 ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) = ∅)
2218, 21impbid1 224 1 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wne 2932  c0 4314  {csn 4620   × cxp 5664  wf 6529  (class class class)co 7401  m cmap 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818
This theorem is referenced by:  map0  8877  mapdom2  9144  map0cor  47709
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