Theorem map0e 7781

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	⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1𝑜)

Proof of Theorem map0e

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4718	. . . 4 ⊢ ∅ ∈ V
2		elmapg 7757	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
3	1, 2	mpan2 703	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
4		f0bi 6001	. . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅)
5		el1o 7466	. . . 4 ⊢ (𝑓 ∈ 1𝑜 ↔ 𝑓 = ∅)
6	4, 5	bitr4i 266	. . 3 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 ∈ 1𝑜)
7	3, 6	syl6bb 275	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓 ∈ 1𝑜))
8	7	eqrdv 2608	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1𝑜)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ⟶wf 5800  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-map 7746

This theorem is referenced by:  fseqenlem1  8730  infmap2  8923  pwcfsdom  9284  cfpwsdom  9285  hashmap  13082  mat0dimbas0  20091  mavmul0  20177  mavmul0g  20178  cramer0  20315  poimirlem28  32607  pwslnmlem0  36679  lincval0  41998  lco0  42010  linds0  42048