Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  map0e Structured version   Visualization version   GIF version

Theorem map0e 7839
 Description: Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
map0e (𝐴𝑉 → (𝐴𝑚 ∅) = 1𝑜)

Proof of Theorem map0e
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 0ex 4750 . . . 4 ∅ ∈ V
2 elmapg 7815 . . . 4 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
31, 2mpan2 706 . . 3 (𝐴𝑉 → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
4 f0bi 6045 . . . 4 (𝑓:∅⟶𝐴𝑓 = ∅)
5 el1o 7524 . . . 4 (𝑓 ∈ 1𝑜𝑓 = ∅)
64, 5bitr4i 267 . . 3 (𝑓:∅⟶𝐴𝑓 ∈ 1𝑜)
73, 6syl6bb 276 . 2 (𝐴𝑉 → (𝑓 ∈ (𝐴𝑚 ∅) ↔ 𝑓 ∈ 1𝑜))
87eqrdv 2619 1 (𝐴𝑉 → (𝐴𝑚 ∅) = 1𝑜)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∅c0 3891  ⟶wf 5843  (class class class)co 6604  1𝑜c1o 7498   ↑𝑚 cmap 7802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1o 7505  df-map 7804 This theorem is referenced by:  fseqenlem1  8791  infmap2  8984  pwcfsdom  9349  cfpwsdom  9350  hashmap  13162  mat0dimbas0  20191  mavmul0  20277  mavmul0g  20278  cramer0  20415  poimirlem28  33066  pwslnmlem0  37138  lincval0  41489  lco0  41501  linds0  41539
 Copyright terms: Public domain W3C validator