Theorem pwen 8133

Description: If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)

Assertion

Ref	Expression
pwen	⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)

Proof of Theorem pwen

Step	Hyp	Ref	Expression
1		relen 7960	. . . 4 ⊢ Rel ≈
2	1	brrelexi 5158	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
3		pw2eng 8066	. . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴))
5		2onn 7720	. . . . . 6 ⊢ 2𝑜 ∈ ω
6	5	elexi 3213	. . . . 5 ⊢ 2𝑜 ∈ V
7	6	enref 7988	. . . 4 ⊢ 2𝑜 ≈ 2𝑜
8		mapen 8124	. . . 4 ⊢ ((2𝑜 ≈ 2𝑜 ∧ 𝐴 ≈ 𝐵) → (2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵))
9	7, 8	mpan 706	. . 3 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵))
10	1	brrelex2i 5159	. . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
11		pw2eng 8066	. . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
12		ensym 8005	. . . 4 ⊢ (𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵) → (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵)
13	10, 11, 12	3syl 18	. . 3 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵)
14		entr 8008	. . 3 ⊢ (((2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵) ∧ (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵)
15	9, 13, 14	syl2anc 693	. 2 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵)
16		entr 8008	. 2 ⊢ ((𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴) ∧ (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵)
17	4, 15, 16	syl2anc 693	1 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1990  Vcvv 3200  𝒫 cpw 4158   class class class wbr 4653  (class class class)co 6650  ωcom 7065  2_𝑜c2o 7554   ↑_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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  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-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956

This theorem is referenced by:  pwfi  8261  dfac12k  8969  pwcdaidm  9017  pwsdompw  9026  ackbij2lem2  9062  engch  9450  gchdomtri  9451  canthp1lem1  9474  gchcdaidm  9490  gchxpidm  9491  gchpwdom  9492  gchhar  9501  inar1  9597  rexpen  14957  enrelmap  38291  enrelmapr  38292