Theorem map1 8036

Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)

Assertion

Ref	Expression
map1	⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ≈ 1𝑜)

Proof of Theorem map1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 6680	. 2 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ∈ V)
2		df1o2 7572	. . . 4 ⊢ 1𝑜 = {∅}
3		p0ex 4853	. . . 4 ⊢ {∅} ∈ V
4	2, 3	eqeltri 2697	. . 3 ⊢ 1𝑜 ∈ V
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → 1𝑜 ∈ V)
6		0ex 4790	. . 3 ⊢ ∅ ∈ V
7	6	2a1i 12	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) → ∅ ∈ V))
8		xpexg 6960	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
9	3, 8	mpan2 707	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V)
10	9	a1d 25	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
11		el1o 7579	. . . . 5 ⊢ (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅)
12	11	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅))
13	2	oveq1i 6660	. . . . . . 7 ⊢ (1𝑜 ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
14	13	eleq2i 2693	. . . . . 6 ⊢ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
15		elmapg 7870	. . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
16	3, 15	mpan 706	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
17	14, 16	syl5bb 272	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
18	6	fconst2 6470	. . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
19	17, 18	syl6rbb 277	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)))
20	12, 19	anbi12d 747	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴))))
21		ancom 466	. . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅))
22	20, 21	syl6rbb 277	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅}))))
23	1, 5, 7, 10, 22	en2d 7991	1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ≈ 1𝑜)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177   class class class wbr 4653   × cxp 5112  ⟶wf 5884  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857   ≈ cen 7952

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-map 7859  df-en 7956

This theorem is referenced by: (None)