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Theorem map0g 8136
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 4131 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6309 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 8108 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3syl5ibr 238 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵)))
5 ne0i 4121 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
76exlimdv 2029 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
81, 7syl5bi 234 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴𝑚 𝐵) ≠ ∅))
98necon4d 2995 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6301 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 6238 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 250 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 8108 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴𝑚 𝐵) ↔ ∅:𝐵𝐴))
1412, 13syl5ibr 238 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴𝑚 𝐵)))
15 ne0i 4121 . . . . 5 (∅ ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴𝑚 𝐵) ≠ ∅))
1716necon2d 2994 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 509 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 6885 . . 3 (𝐴 = ∅ → (𝐴𝑚 𝐵) = (∅ ↑𝑚 𝐵))
20 map0b 8135 . . 3 (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅)
2119, 20sylan9eq 2853 . 2 ((𝐴 = ∅ ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) = ∅)
2218, 21impbid1 217 1 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wne 2971  c0 4115  {csn 4368   × cxp 5310  wf 6097  (class class class)co 6878  𝑚 cmap 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097
This theorem is referenced by:  map0  8138  mapdom2  8373
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