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Theorem map0b 7782
Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
map0b (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)

Proof of Theorem map0b
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elmapi 7765 . . . 4 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝑓:𝐴⟶∅)
2 fdm 5964 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴)
3 frn 5966 . . . . . . 7 (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅)
4 ss0 3926 . . . . . . 7 (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅)
53, 4syl 17 . . . . . 6 (𝑓:𝐴⟶∅ → ran 𝑓 = ∅)
6 dm0rn0 5263 . . . . . 6 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
75, 6sylibr 223 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = ∅)
82, 7eqtr3d 2646 . . . 4 (𝑓:𝐴⟶∅ → 𝐴 = ∅)
91, 8syl 17 . . 3 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝐴 = ∅)
109necon3ai 2807 . 2 (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑𝑚 𝐴))
1110eq0rdv 3931 1 (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  wne 2780  wss 3540  c0 3874  dom cdm 5038  ran crn 5039  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:  map0g  7783  mapdom2  8016  ply1plusgfvi  19433
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