Theorem pw2eng 8308

Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2_𝑜. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)

Assertion

Ref	Expression
pw2eng	⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴))

Proof of Theorem pw2eng

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

Step	Hyp	Ref	Expression
1		pwexg 5048	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		ovexd 6912	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, {∅}} ↑𝑚 𝐴) ∈ V)
3		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0ex 4984	. . . . 5 ⊢ ∅ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
6		p0ex 5053	. . . . 5 ⊢ {∅} ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
8		0nep0 5028	. . . . 5 ⊢ ∅ ≠ {∅}
9	8	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ≠ {∅})
10		eqid 2799	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅)))
11	3, 5, 7, 9, 10	pw2f1o 8307	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑𝑚 𝐴))
12		f1oen2g 8212	. . 3 ⊢ ((𝒫 𝐴 ∈ V ∧ ({∅, {∅}} ↑𝑚 𝐴) ∈ V ∧ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑𝑚 𝐴)) → 𝒫 𝐴 ≈ ({∅, {∅}} ↑𝑚 𝐴))
13	1, 2, 11, 12	syl3anc 1491	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑𝑚 𝐴))
14		df2o2 7814	. . 3 ⊢ 2𝑜 = {∅, {∅}}
15	14	oveq1i 6888	. 2 ⊢ (2𝑜 ↑𝑚 𝐴) = ({∅, {∅}} ↑𝑚 𝐴)
16	13, 15	syl6breqr 4885	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2157   ≠ wne 2971  Vcvv 3385  ∅c0 4115  ifcif 4277  𝒫 cpw 4349  {csn 4368  {cpr 4370   class class class wbr 4843   ↦ cmpt 4922  –1-1-onto→wf1o 6100  (class class class)co 6878  2_𝑜c2o 7793   ↑_𝑚 cmap 8095   ≈ cen 8192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1o 7799  df-2o 7800  df-map 8097  df-en 8196

This theorem is referenced by:  pw2en  8309  pwen  8375  mappwen  9221  pwcdaen  9295  ackbij1lem5  9334  hauspwdom  21633  enrelmap  39069