Theorem map1 7899

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		ovex 6555	. . 3 ⊢ (1𝑜 ↑𝑚 𝐴) ∈ V
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ∈ V)
3		df1o2 7437	. . . 4 ⊢ 1𝑜 = {∅}
4		p0ex 4774	. . . 4 ⊢ {∅} ∈ V
5	3, 4	eqeltri 2684	. . 3 ⊢ 1𝑜 ∈ V
6	5	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → 1𝑜 ∈ V)
7		0ex 4713	. . 3 ⊢ ∅ ∈ V
8	7	2a1i 12	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) → ∅ ∈ V))
9		xpexg 6836	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
10	4, 9	mpan2 703	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V)
11	10	a1d 25	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
12		el1o 7444	. . . . 5 ⊢ (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅)
13	12	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅))
14	3	oveq1i 6537	. . . . . . 7 ⊢ (1𝑜 ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
15	14	eleq2i 2680	. . . . . 6 ⊢ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
16		elmapg 7735	. . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
17	4, 16	mpan 702	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
18	15, 17	syl5bb 271	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
19	7	fconst2 6353	. . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
20	18, 19	syl6rbb 276	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)))
21	13, 20	anbi12d 743	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴))))
22		ancom 465	. . 3 ⊢ ((𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)) ↔ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅))
23	21, 22	syl6rbb 276	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ∧ 𝑦 = ∅) ↔ (𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅}))))
24	2, 6, 8, 11, 23	en2d 7855	1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ≈ 1𝑜)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125   class class class wbr 4578   × cxp 5026  ⟶wf 5786  (class class class)co 6527  1_𝑜c1o 7418   ↑_𝑚 cmap 7722   ≈ cen 7816

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 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825

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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1o 7425  df-map 7724  df-en 7820

This theorem is referenced by: (None)