Theorem enpw1pw 6075

Description: Unit power class and power class commute within equivalence. 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	⊢ ℘1℘A ≈ ℘℘1A

Proof of Theorem enpw1pw

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

Step	Hyp	Ref	Expression
1		pw1fnf1o 5855	. . . . 5 ⊢ Pw1Fn :1c–1-1-onto→℘1c
2		f1of1 5286	. . . . 5 ⊢ ( Pw1Fn :1c–1-1-onto→℘1c → Pw1Fn :1c–1-1→℘1c)
3	1, 2	ax-mp 5	. . . 4 ⊢ Pw1Fn :1c–1-1→℘1c
4		pw1ss1c 4158	. . . 4 ⊢ ℘1℘A ⊆ 1c
5		f1ores 5300	. . . 4 ⊢ (( Pw1Fn :1c–1-1→℘1c ∧ ℘1℘A ⊆ 1c) → ( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→( Pw1Fn “ ℘1℘A))
6	3, 4, 5	mp2an 653	. . 3 ⊢ ( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→( Pw1Fn “ ℘1℘A)
7		df-ima 4727	. . . . 5 ⊢ ( Pw1Fn “ ℘1℘A) = {x ∣ ∃y ∈ ℘1 ℘Ay Pw1Fn x}
8		vex 2862	. . . . . . . . 9 ⊢ x ∈ V
9	8	elpw 3728	. . . . . . . 8 ⊢ (x ∈ ℘℘1A ↔ x ⊆ ℘1A)
10	8	sspw1 4335	. . . . . . . 8 ⊢ (x ⊆ ℘1A ↔ ∃z(z ⊆ A ∧ x = ℘1z))
11		df-rex 2620	. . . . . . . . 9 ⊢ (∃z ∈ ℘ Ax = ℘1z ↔ ∃z(z ∈ ℘A ∧ x = ℘1z))
12		df-pw 3724	. . . . . . . . . . . 12 ⊢ ℘A = {z ∣ z ⊆ A}
13	12	abeq2i 2460	. . . . . . . . . . 11 ⊢ (z ∈ ℘A ↔ z ⊆ A)
14	13	anbi1i 676	. . . . . . . . . 10 ⊢ ((z ∈ ℘A ∧ x = ℘1z) ↔ (z ⊆ A ∧ x = ℘1z))
15	14	exbii 1582	. . . . . . . . 9 ⊢ (∃z(z ∈ ℘A ∧ x = ℘1z) ↔ ∃z(z ⊆ A ∧ x = ℘1z))
16	11, 15	bitr2i 241	. . . . . . . 8 ⊢ (∃z(z ⊆ A ∧ x = ℘1z) ↔ ∃z ∈ ℘ Ax = ℘1z)
17	9, 10, 16	3bitri 262	. . . . . . 7 ⊢ (x ∈ ℘℘1A ↔ ∃z ∈ ℘ Ax = ℘1z)
18		df-rex 2620	. . . . . . . 8 ⊢ (∃y ∈ ℘1 ℘Ay Pw1Fn x ↔ ∃y(y ∈ ℘1℘A ∧ y Pw1Fn x))
19		elpw1 4144	. . . . . . . . . . 11 ⊢ (y ∈ ℘1℘A ↔ ∃z ∈ ℘ Ay = {z})
20	19	anbi1i 676	. . . . . . . . . 10 ⊢ ((y ∈ ℘1℘A ∧ y Pw1Fn x) ↔ (∃z ∈ ℘ Ay = {z} ∧ y Pw1Fn x))
21		r19.41v 2764	. . . . . . . . . 10 ⊢ (∃z ∈ ℘ A(y = {z} ∧ y Pw1Fn x) ↔ (∃z ∈ ℘ Ay = {z} ∧ y Pw1Fn x))
22	20, 21	bitr4i 243	. . . . . . . . 9 ⊢ ((y ∈ ℘1℘A ∧ y Pw1Fn x) ↔ ∃z ∈ ℘ A(y = {z} ∧ y Pw1Fn x))
23	22	exbii 1582	. . . . . . . 8 ⊢ (∃y(y ∈ ℘1℘A ∧ y Pw1Fn x) ↔ ∃y∃z ∈ ℘ A(y = {z} ∧ y Pw1Fn x))
24		rexcom4 2878	. . . . . . . . 9 ⊢ (∃z ∈ ℘ A∃y(y = {z} ∧ y Pw1Fn x) ↔ ∃y∃z ∈ ℘ A(y = {z} ∧ y Pw1Fn x))
25		snex 4111	. . . . . . . . . . . 12 ⊢ {z} ∈ V
26		breq1 4642	. . . . . . . . . . . 12 ⊢ (y = {z} → (y Pw1Fn x ↔ {z} Pw1Fn x))
27	25, 26	ceqsexv 2894	. . . . . . . . . . 11 ⊢ (∃y(y = {z} ∧ y Pw1Fn x) ↔ {z} Pw1Fn x)
28		vex 2862	. . . . . . . . . . . 12 ⊢ z ∈ V
29	28	brpw1fn 5854	. . . . . . . . . . 11 ⊢ ({z} Pw1Fn x ↔ x = ℘1z)
30	27, 29	bitri 240	. . . . . . . . . 10 ⊢ (∃y(y = {z} ∧ y Pw1Fn x) ↔ x = ℘1z)
31	30	rexbii 2639	. . . . . . . . 9 ⊢ (∃z ∈ ℘ A∃y(y = {z} ∧ y Pw1Fn x) ↔ ∃z ∈ ℘ Ax = ℘1z)
32	24, 31	bitr3i 242	. . . . . . . 8 ⊢ (∃y∃z ∈ ℘ A(y = {z} ∧ y Pw1Fn x) ↔ ∃z ∈ ℘ Ax = ℘1z)
33	18, 23, 32	3bitri 262	. . . . . . 7 ⊢ (∃y ∈ ℘1 ℘Ay Pw1Fn x ↔ ∃z ∈ ℘ Ax = ℘1z)
34	17, 33	bitr4i 243	. . . . . 6 ⊢ (x ∈ ℘℘1A ↔ ∃y ∈ ℘1 ℘Ay Pw1Fn x)
35	34	abbi2i 2464	. . . . 5 ⊢ ℘℘1A = {x ∣ ∃y ∈ ℘1 ℘Ay Pw1Fn x}
36	7, 35	eqtr4i 2376	. . . 4 ⊢ ( Pw1Fn “ ℘1℘A) = ℘℘1A
37		f1oeq3 5283	. . . 4 ⊢ (( Pw1Fn “ ℘1℘A) = ℘℘1A → (( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→( Pw1Fn “ ℘1℘A) ↔ ( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→℘℘1A))
38	36, 37	ax-mp 5	. . 3 ⊢ (( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→( Pw1Fn “ ℘1℘A) ↔ ( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→℘℘1A)
39	6, 38	mpbi 199	. 2 ⊢ ( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→℘℘1A
40		pw1fnex 5852	. . . 4 ⊢ Pw1Fn ∈ V
41		enpw1pw.1	. . . . . 6 ⊢ A ∈ V
42	41	pwex 4329	. . . . 5 ⊢ ℘A ∈ V
43	42	pw1ex 4303	. . . 4 ⊢ ℘1℘A ∈ V
44	40, 43	resex 5117	. . 3 ⊢ ( Pw1Fn ↾ ℘1℘A) ∈ V
45	44	f1oen 6033	. 2 ⊢ (( Pw1Fn ↾ ℘1℘A):℘1℘A–1-1-onto→℘℘1A → ℘1℘A ≈ ℘℘1A)
46	39, 45	ax-mp 5	1 ⊢ ℘1℘A ≈ ℘℘1A

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  {csn 3737  1_cc1c 4134  ℘₁cpw1 4135   class class class wbr 4639   “ cima 4722   ↾ cres 4774  –1-1→wf1 4778  –1-1-onto→wf1o 4780   Pw1Fn cpw1fn 5765   ≈ cen 6028

This theorem was proved from 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752  df-pw1fn 5766  df-en 6029

This theorem is referenced by:  ncpwpw1  6153