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Theorem map0g 7844
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 3909 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑓 𝑓𝐴)
2 fconst6g 6053 . . . . . . . 8 (𝑓𝐴 → (𝐵 × {𝑓}):𝐵𝐴)
3 elmapg 7818 . . . . . . . 8 ((𝐴𝑉𝐵𝑊) → ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) ↔ (𝐵 × {𝑓}):𝐵𝐴))
42, 3syl5ibr 236 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵)))
5 ne0i 3899 . . . . . . 7 ((𝐵 × {𝑓}) ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
64, 5syl6 35 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
76exlimdv 1858 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∃𝑓 𝑓𝐴 → (𝐴𝑚 𝐵) ≠ ∅))
81, 7syl5bi 232 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 ≠ ∅ → (𝐴𝑚 𝐵) ≠ ∅))
98necon4d 2814 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐴 = ∅))
10 f0 6045 . . . . . . 7 ∅:∅⟶𝐴
11 feq2 5986 . . . . . . 7 (𝐵 = ∅ → (∅:𝐵𝐴 ↔ ∅:∅⟶𝐴))
1210, 11mpbiri 248 . . . . . 6 (𝐵 = ∅ → ∅:𝐵𝐴)
13 elmapg 7818 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (∅ ∈ (𝐴𝑚 𝐵) ↔ ∅:𝐵𝐴))
1412, 13syl5ibr 236 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → ∅ ∈ (𝐴𝑚 𝐵)))
15 ne0i 3899 . . . . 5 (∅ ∈ (𝐴𝑚 𝐵) → (𝐴𝑚 𝐵) ≠ ∅)
1614, 15syl6 35 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 = ∅ → (𝐴𝑚 𝐵) ≠ ∅))
1716necon2d 2813 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → 𝐵 ≠ ∅))
189, 17jcad 555 . 2 ((𝐴𝑉𝐵𝑊) → ((𝐴𝑚 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 ≠ ∅)))
19 oveq1 6614 . . 3 (𝐴 = ∅ → (𝐴𝑚 𝐵) = (∅ ↑𝑚 𝐵))
20 map0b 7843 . . 3 (𝐵 ≠ ∅ → (∅ ↑𝑚 𝐵) = ∅)
2119, 20sylan9eq 2675 . 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 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  c0 3893  {csn 4150   × cxp 5074  wf 5845  (class class class)co 6607  𝑚 cmap 7805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807
This theorem is referenced by:  map0  7845  mapdom2  8078
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