Theorem map1 8203

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 6844	. 2 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ∈ V)
2		df1o2 7743	. . . 4 ⊢ 1𝑜 = {∅}
3		p0ex 5002	. . . 4 ⊢ {∅} ∈ V
4	2, 3	eqeltri 2835	. . 3 ⊢ 1𝑜 ∈ V
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → 1𝑜 ∈ V)
6		0ex 4942	. . 3 ⊢ ∅ ∈ V
7	6	2a1i 12	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) → ∅ ∈ V))
8		xpexg 7126	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {∅} ∈ V) → (𝐴 × {∅}) ∈ V)
9	3, 8	mpan2 709	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {∅}) ∈ V)
10	9	a1d 25	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 → (𝐴 × {∅}) ∈ V))
11		el1o 7750	. . . . 5 ⊢ (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅)
12	11	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 1𝑜 ↔ 𝑦 = ∅))
13	2	oveq1i 6824	. . . . . . 7 ⊢ (1𝑜 ↑𝑚 𝐴) = ({∅} ↑𝑚 𝐴)
14	13	eleq2i 2831	. . . . . 6 ⊢ (𝑥 ∈ (1𝑜 ↑𝑚 𝐴) ↔ 𝑥 ∈ ({∅} ↑𝑚 𝐴))
15		elmapg 8038	. . . . . . 7 ⊢ (({∅} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ ({∅} ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶{∅}))
16	3, 15	mpan 708	. . . . . 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 6635	. . . . 5 ⊢ (𝑥:𝐴⟶{∅} ↔ 𝑥 = (𝐴 × {∅}))
19	17, 18	syl6rbb 277	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 = (𝐴 × {∅}) ↔ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴)))
20	12, 19	anbi12d 749	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑦 ∈ 1𝑜 ∧ 𝑥 = (𝐴 × {∅})) ↔ (𝑦 = ∅ ∧ 𝑥 ∈ (1𝑜 ↑𝑚 𝐴))))
21		ancom 465	. . 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 8159	1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ↑𝑚 𝐴) ≈ 1𝑜)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ∅c0 4058  {csn 4321   class class class wbr 4804   × cxp 5264  ⟶wf 6045  (class class class)co 6814  1_𝑜c1o 7723   ↑_𝑚 cmap 8025   ≈ cen 8120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1o 7730  df-map 8027  df-en 8124

This theorem is referenced by: (None)