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Theorem map0b 8874
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 β‰  βˆ… β†’ (βˆ… ↑m 𝐴) = βˆ…)

Proof of Theorem map0b
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elmapi 8840 . . . 4 (𝑓 ∈ (βˆ… ↑m 𝐴) β†’ 𝑓:π΄βŸΆβˆ…)
2 fdm 6717 . . . . 5 (𝑓:π΄βŸΆβˆ… β†’ dom 𝑓 = 𝐴)
3 frn 6715 . . . . . . 7 (𝑓:π΄βŸΆβˆ… β†’ ran 𝑓 βŠ† βˆ…)
4 ss0 4391 . . . . . . 7 (ran 𝑓 βŠ† βˆ… β†’ ran 𝑓 = βˆ…)
53, 4syl 17 . . . . . 6 (𝑓:π΄βŸΆβˆ… β†’ ran 𝑓 = βˆ…)
6 dm0rn0 5915 . . . . . 6 (dom 𝑓 = βˆ… ↔ ran 𝑓 = βˆ…)
75, 6sylibr 233 . . . . 5 (𝑓:π΄βŸΆβˆ… β†’ dom 𝑓 = βˆ…)
82, 7eqtr3d 2766 . . . 4 (𝑓:π΄βŸΆβˆ… β†’ 𝐴 = βˆ…)
91, 8syl 17 . . 3 (𝑓 ∈ (βˆ… ↑m 𝐴) β†’ 𝐴 = βˆ…)
109necon3ai 2957 . 2 (𝐴 β‰  βˆ… β†’ Β¬ 𝑓 ∈ (βˆ… ↑m 𝐴))
1110eq0rdv 4397 1 (𝐴 β‰  βˆ… β†’ (βˆ… ↑m 𝐴) = βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   β‰  wne 2932   βŠ† wss 3941  βˆ…c0 4315  dom cdm 5667  ran crn 5668  βŸΆwf 6530  (class class class)co 7402   ↑m cmap 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819
This theorem is referenced by:  map0g  8875  mapdom2  9145  ply1plusgfvi  22104  satf0  34880  prv0  34938
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