Theorem pwen 8249

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 8077	. . . 4 ⊢ Rel ≈
2	1	brrelexi 5267	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
3		pw2eng 8182	. . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴))
4	2, 3	syl 17	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴))
5		2onn 7840	. . . . . 6 ⊢ 2𝑜 ∈ ω
6	5	elexi 3317	. . . . 5 ⊢ 2𝑜 ∈ V
7	6	enref 8105	. . . 4 ⊢ 2𝑜 ≈ 2𝑜
8		mapen 8240	. . . 4 ⊢ ((2𝑜 ≈ 2𝑜 ∧ 𝐴 ≈ 𝐵) → (2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵))
9	7, 8	mpan 708	. . 3 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵))
10	1	brrelex2i 5268	. . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
11		pw2eng 8182	. . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
12		ensym 8121	. . . 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 8124	. . 3 ⊢ (((2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵) ∧ (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵)
15	9, 13, 14	syl2anc 696	. 2 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵)
16		entr 8124	. 2 ⊢ ((𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴) ∧ (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵)
17	4, 15, 16	syl2anc 696	1 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2103  Vcvv 3304  𝒫 cpw 4266   class class class wbr 4760  (class class class)co 6765  ωcom 7182  2_𝑜c2o 7674   ↑_𝑚 cmap 7974   ≈ cen 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-1o 7680  df-2o 7681  df-er 7862  df-map 7976  df-en 8073

This theorem is referenced by:  pwfi  8377  dfac12k  9082  pwcdaidm  9130  pwsdompw  9139  ackbij2lem2  9175  engch  9563  gchdomtri  9564  canthp1lem1  9587  gchcdaidm  9603  gchxpidm  9604  gchpwdom  9605  gchhar  9614  inar1  9710  rexpen  15077  enrelmap  38710  enrelmapr  38711