Theorem enpw1pw 6076

Description: Unit power class and power class commute up to equinumerosity. Theorem XI.1.35 of [Rosser] p. 368. (Contributed by SF, 26-Feb-2015.)

Hypothesis

Ref	Expression
enpw1pw.1	⊢ A ∈ V

Assertion

Ref	Expression
enpw1pw	⊢ 𝒫1𝒫 A ≈ 𝒫 𝒫1A

Proof of Theorem enpw1pw

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

Step	Hyp	Ref	Expression
1		pw1fnf1o 5856	. . . . 5 ⊢ Pw1Fn :1c–1-1-onto→𝒫 1c
2		f1of1 5287	. . . . 5 ⊢ ( Pw1Fn :1c–1-1-onto→𝒫 1c → Pw1Fn :1c–1-1→𝒫 1c)
3	1, 2	ax-mp 5	. . . 4 ⊢ Pw1Fn :1c–1-1→𝒫 1c
4		pw1ss1c 4159	. . . 4 ⊢ 𝒫1𝒫 A ⊆ 1c
5		f1ores 5301	. . . 4 ⊢ (( Pw1Fn :1c–1-1→𝒫 1c ∧ 𝒫1𝒫 A ⊆ 1c) → ( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→( Pw1Fn “ 𝒫1𝒫 A))
6	3, 4, 5	mp2an 653	. . 3 ⊢ ( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→( Pw1Fn “ 𝒫1𝒫 A)
7		df-ima 4728	. . . . 5 ⊢ ( Pw1Fn “ 𝒫1𝒫 A) = {x ∣ ∃y ∈ 𝒫1 𝒫 Ay Pw1Fn x}
8		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
9	8	elpw 3729	. . . . . . . 8 ⊢ (x ∈ 𝒫 𝒫1A ↔ x ⊆ 𝒫1A)
10	8	sspw1 4336	. . . . . . . 8 ⊢ (x ⊆ 𝒫1A ↔ ∃z(z ⊆ A ∧ x = 𝒫1z))
11		df-rex 2621	. . . . . . . . 9 ⊢ (∃z ∈ 𝒫 Ax = 𝒫1z ↔ ∃z(z ∈ 𝒫 A ∧ x = 𝒫1z))
12		df-pw 3725	. . . . . . . . . . . 12 ⊢ 𝒫 A = {z ∣ z ⊆ A}
13	12	eqabri 2461	. . . . . . . . . . 11 ⊢ (z ∈ 𝒫 A ↔ z ⊆ A)
14	13	anbi1i 676	. . . . . . . . . 10 ⊢ ((z ∈ 𝒫 A ∧ x = 𝒫1z) ↔ (z ⊆ A ∧ x = 𝒫1z))
15	14	exbii 1582	. . . . . . . . 9 ⊢ (∃z(z ∈ 𝒫 A ∧ x = 𝒫1z) ↔ ∃z(z ⊆ A ∧ x = 𝒫1z))
16	11, 15	bitr2i 241	. . . . . . . 8 ⊢ (∃z(z ⊆ A ∧ x = 𝒫1z) ↔ ∃z ∈ 𝒫 Ax = 𝒫1z)
17	9, 10, 16	3bitri 262	. . . . . . 7 ⊢ (x ∈ 𝒫 𝒫1A ↔ ∃z ∈ 𝒫 Ax = 𝒫1z)
18		df-rex 2621	. . . . . . . 8 ⊢ (∃y ∈ 𝒫1 𝒫 Ay Pw1Fn x ↔ ∃y(y ∈ 𝒫1𝒫 A ∧ y Pw1Fn x))
19		elpw1 4145	. . . . . . . . . . 11 ⊢ (y ∈ 𝒫1𝒫 A ↔ ∃z ∈ 𝒫 Ay = {z})
20	19	anbi1i 676	. . . . . . . . . 10 ⊢ ((y ∈ 𝒫1𝒫 A ∧ y Pw1Fn x) ↔ (∃z ∈ 𝒫 Ay = {z} ∧ y Pw1Fn x))
21		r19.41v 2765	. . . . . . . . . 10 ⊢ (∃z ∈ 𝒫 A(y = {z} ∧ y Pw1Fn x) ↔ (∃z ∈ 𝒫 Ay = {z} ∧ y Pw1Fn x))
22	20, 21	bitr4i 243	. . . . . . . . 9 ⊢ ((y ∈ 𝒫1𝒫 A ∧ y Pw1Fn x) ↔ ∃z ∈ 𝒫 A(y = {z} ∧ y Pw1Fn x))
23	22	exbii 1582	. . . . . . . 8 ⊢ (∃y(y ∈ 𝒫1𝒫 A ∧ y Pw1Fn x) ↔ ∃y∃z ∈ 𝒫 A(y = {z} ∧ y Pw1Fn x))
24		rexcom4 2879	. . . . . . . . 9 ⊢ (∃z ∈ 𝒫 A∃y(y = {z} ∧ y Pw1Fn x) ↔ ∃y∃z ∈ 𝒫 A(y = {z} ∧ y Pw1Fn x))
25		snex 4112	. . . . . . . . . . . 12 ⊢ {z} ∈ V
26		breq1 4643	. . . . . . . . . . . 12 ⊢ (y = {z} → (y Pw1Fn x ↔ {z} Pw1Fn x))
27	25, 26	ceqsexv 2895	. . . . . . . . . . 11 ⊢ (∃y(y = {z} ∧ y Pw1Fn x) ↔ {z} Pw1Fn x)
28		vex 2863	. . . . . . . . . . . 12 ⊢ z ∈ V
29	28	brpw1fn 5855	. . . . . . . . . . 11 ⊢ ({z} Pw1Fn x ↔ x = 𝒫1z)
30	27, 29	bitri 240	. . . . . . . . . 10 ⊢ (∃y(y = {z} ∧ y Pw1Fn x) ↔ x = 𝒫1z)
31	30	rexbii 2640	. . . . . . . . 9 ⊢ (∃z ∈ 𝒫 A∃y(y = {z} ∧ y Pw1Fn x) ↔ ∃z ∈ 𝒫 Ax = 𝒫1z)
32	24, 31	bitr3i 242	. . . . . . . 8 ⊢ (∃y∃z ∈ 𝒫 A(y = {z} ∧ y Pw1Fn x) ↔ ∃z ∈ 𝒫 Ax = 𝒫1z)
33	18, 23, 32	3bitri 262	. . . . . . 7 ⊢ (∃y ∈ 𝒫1 𝒫 Ay Pw1Fn x ↔ ∃z ∈ 𝒫 Ax = 𝒫1z)
34	17, 33	bitr4i 243	. . . . . 6 ⊢ (x ∈ 𝒫 𝒫1A ↔ ∃y ∈ 𝒫1 𝒫 Ay Pw1Fn x)
35	34	eqabi 2465	. . . . 5 ⊢ 𝒫 𝒫1A = {x ∣ ∃y ∈ 𝒫1 𝒫 Ay Pw1Fn x}
36	7, 35	eqtr4i 2376	. . . 4 ⊢ ( Pw1Fn “ 𝒫1𝒫 A) = 𝒫 𝒫1A
37		f1oeq3 5284	. . . 4 ⊢ (( Pw1Fn “ 𝒫1𝒫 A) = 𝒫 𝒫1A → (( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→( Pw1Fn “ 𝒫1𝒫 A) ↔ ( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→𝒫 𝒫1A))
38	36, 37	ax-mp 5	. . 3 ⊢ (( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→( Pw1Fn “ 𝒫1𝒫 A) ↔ ( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→𝒫 𝒫1A)
39	6, 38	mpbi 199	. 2 ⊢ ( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→𝒫 𝒫1A
40		pw1fnex 5853	. . . 4 ⊢ Pw1Fn ∈ V
41		enpw1pw.1	. . . . . 6 ⊢ A ∈ V
42	41	pwex 4330	. . . . 5 ⊢ 𝒫 A ∈ V
43	42	pw1ex 4304	. . . 4 ⊢ 𝒫1𝒫 A ∈ V
44	40, 43	resex 5118	. . 3 ⊢ ( Pw1Fn ↾ 𝒫1𝒫 A) ∈ V
45	44	f1oen 6034	. 2 ⊢ (( Pw1Fn ↾ 𝒫1𝒫 A):𝒫1𝒫 A–1-1-onto→𝒫 𝒫1A → 𝒫1𝒫 A ≈ 𝒫 𝒫1A)
46	39, 45	ax-mp 5	1 ⊢ 𝒫1𝒫 A ≈ 𝒫 𝒫1A

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  𝒫 cpw 3723  {csn 3738  1_cc1c 4135  𝒫₁cpw1 4136   class class class wbr 4640   “ cima 4723   ↾ cres 4775  –1-1→wf1 4779  –1-1-onto→wf1o 4781   Pw1Fn cpw1fn 5766   ≈ cen 6029

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753  df-pw1fn 5767  df-en 6030

This theorem is used by:  ncpwpw1  6154