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Theorem map0b 6547
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 6530 . . . 4 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝑓:𝐴⟶∅)
2 fdm 5246 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴)
3 frn 5249 . . . . . . 7 (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅)
4 ss0 3371 . . . . . . 7 (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅)
53, 4syl 14 . . . . . 6 (𝑓:𝐴⟶∅ → ran 𝑓 = ∅)
6 dm0rn0 4724 . . . . . 6 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
75, 6sylibr 133 . . . . 5 (𝑓:𝐴⟶∅ → dom 𝑓 = ∅)
82, 7eqtr3d 2150 . . . 4 (𝑓:𝐴⟶∅ → 𝐴 = ∅)
91, 8syl 14 . . 3 (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝐴 = ∅)
109necon3ai 2332 . 2 (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑𝑚 𝐴))
1110eq0rdv 3375 1 (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  wne 2283  wss 3039  c0 3331  dom cdm 4507  ran crn 4508  wf 5087  (class class class)co 5740  𝑚 cmap 6508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-map 6510
This theorem is referenced by:  map0g  6548
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