Theorem pw2eng 8626

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 5282	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		ovexd 7194	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({∅, {∅}} ↑m 𝐴) ∈ V)
3		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
4		0ex 5214	. . . . 5 ⊢ ∅ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ∈ V)
6		p0ex 5288	. . . . 5 ⊢ {∅} ∈ V
7	6	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {∅} ∈ V)
8		0nep0 5261	. . . . 5 ⊢ ∅ ≠ {∅}
9	8	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∅ ≠ {∅})
10		eqid 2824	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅)))
11	3, 5, 7, 9, 10	pw2f1o 8625	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑥, {∅}, ∅))):𝒫 𝐴–1-1-onto→({∅, {∅}} ↑m 𝐴))
12		f1oen2g 8529	. . 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 1367	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ ({∅, {∅}} ↑m 𝐴))
14		df2o2 8121	. . 3 ⊢ 2o = {∅, {∅}}
15	14	oveq1i 7169	. 2 ⊢ (2o ↑m 𝐴) = ({∅, {∅}} ↑m 𝐴)
16	13, 15	breqtrrdi 5111	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ≈ (2o ↑m 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2113   ≠ wne 3019  Vcvv 3497  ∅c0 4294  ifcif 4470  𝒫 cpw 4542  {csn 4570  {cpr 4572   class class class wbr 5069   ↦ cmpt 5149  –1-1-onto→wf1o 6357  (class class class)co 7159  2_oc2o 8099   ↑_m cmap 8409   ≈ cen 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1o 8105  df-2o 8106  df-map 8411  df-en 8513

This theorem is referenced by:  pw2en  8627  pwen  8693  mappwen  9541  pwdjuen  9610  ackbij1lem5  9649  hauspwdom  22112  enrelmap  40349