Theorem map0e 7895

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 4790	. . . 4 ⊢ ∅ ∈ V
2		elmapg 7870	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
3	1, 2	mpan2 707	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓:∅⟶𝐴))
4		f0bi 6088	. . . 4 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 = ∅)
5		el1o 7579	. . . 4 ⊢ (𝑓 ∈ 1𝑜 ↔ 𝑓 = ∅)
6	4, 5	bitr4i 267	. . 3 ⊢ (𝑓:∅⟶𝐴 ↔ 𝑓 ∈ 1𝑜)
7	3, 6	syl6bb 276	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (𝐴 ↑𝑚 ∅) ↔ 𝑓 ∈ 1𝑜))
8	7	eqrdv 2620	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) = 1𝑜)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  ⟶wf 5884  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-map 7859

This theorem is referenced by:  fseqenlem1  8847  infmap2  9040  pwcfsdom  9405  cfpwsdom  9406  hashmap  13222  mat0dimbas0  20272  mavmul0  20358  mavmul0g  20359  cramer0  20496  poimirlem28  33437  pwslnmlem0  37661  lincval0  42204  lco0  42216  linds0  42254