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Theorem map0g 8825
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 4307 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6732 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8781 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3imbitrrid 245 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴m 𝐵)))
5 ne0i 4295 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
76exlimdv 1937 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴m 𝐵) ≠ ∅))
81, 7biimtrid 241 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴m 𝐵) ≠ ∅))
98necon4d 2964 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6724 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6651 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 258 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8781 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴m 𝐵) ↔ ∅:𝐵𝐴))
1412, 13imbitrrid 245 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴m 𝐵)))
15 ne0i 4295 . . . . 5 (∅ ∈ (𝐴m 𝐵) → (𝐴m 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴m 𝐵) ≠ ∅))
1716necon2d 2963 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 514 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴m 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 7365 . . 3 (𝐴 = ∅ → (𝐴m 𝐵) = (∅ ↑m 𝐵))
20 map0b 8824 . . 3 (𝐵 ≠ ∅ → (∅ ↑m 𝐵) = ∅)
2119, 20sylan9eq 2793 . 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 397   = wceq 1542  wex 1782  wcel 2107  wne 2940  c0 4283  {csn 4587   × cxp 5632  wf 6493  (class class class)co 7358  m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770
This theorem is referenced by:  map0  8828  mapdom2  9095  map0cor  47007
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