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Theorem mapsnen 8988
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 8987 . 2 (𝐴 ∈ V → (𝐴m {𝐵}) ≈ 𝐴)
61, 5ax-mp 5 1 (𝐴m {𝐵}) ≈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3448  {csn 4591   class class class wbr 5110  (class class class)co 7362  m cmap 8772  cen 8887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-en 8891
This theorem is referenced by: (None)
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