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Theorem mapsnen 9036
Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 17-Jul-2022.)
Hypotheses
Ref Expression
mapsnen.1 𝐴 ∈ V
mapsnen.2 𝐵 ∈ V
Assertion
Ref Expression
mapsnen (𝐴m {𝐵}) ≈ 𝐴

Proof of Theorem mapsnen
StepHypRef Expression
1 mapsnen.1 . 2 𝐴 ∈ V
2 id 22 . . 3 (𝐴 ∈ V → 𝐴 ∈ V)
3 mapsnen.2 . . . 4 𝐵 ∈ V
43a1i 11 . . 3 (𝐴 ∈ V → 𝐵 ∈ V)
52, 4mapsnend 9035 . 2 (𝐴 ∈ V → (𝐴m {𝐵}) ≈ 𝐴)
61, 5ax-mp 5 1 (𝐴m {𝐵}) ≈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3468  {csn 4623   class class class wbr 5141  (class class class)co 7404  m cmap 8819  cen 8935
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-en 8939
This theorem is referenced by: (None)
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