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

Theorem mapsnen 8308
 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 8307 . 2 (𝐴 ∈ V → (𝐴𝑚 {𝐵}) ≈ 𝐴)
61, 5ax-mp 5 1 (𝐴𝑚 {𝐵}) ≈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2164  Vcvv 3414  {csn 4399   class class class wbr 4875  (class class class)co 6910   ↑𝑚 cmap 8127   ≈ cen 8225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-en 8229 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator