MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  map0b Structured version   Visualization version   GIF version

Theorem map0b 8493
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 8459 . . . 4 (𝑓 ∈ (∅ ↑m 𝐴) → 𝑓:𝐴⟶∅)
2 fdm 6513 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴)
3 frn 6511 . . . . . . 7 (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅)
4 ss0 4287 . . . . . . 7 (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅)
53, 4syl 17 . . . . . 6 (𝑓:𝐴⟶∅ → ran 𝑓 = ∅)
6 dm0rn0 5768 . . . . . 6 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
75, 6sylibr 237 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = ∅)
82, 7eqtr3d 2775 . . . 4 (𝑓:𝐴⟶∅ → 𝐴 = ∅)
91, 8syl 17 . . 3 (𝑓 ∈ (∅ ↑m 𝐴) → 𝐴 = ∅)
109necon3ai 2959 . 2 (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑m 𝐴))
1110eq0rdv 4293 1 (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wne 2934  wss 3843  c0 4211  dom cdm 5525  ran crn 5526  wf 6335  (class class class)co 7170  m cmap 8437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-map 8439
This theorem is referenced by:  map0g  8494  mapdom2  8738  ply1plusgfvi  21017  satf0  32905  prv0  32963
  Copyright terms: Public domain W3C validator