Theorem enpw1 6063

Description: Two classes are equinumerous iff their unit power classes are equinumerous. Theorem XI.1.33 of [Rosser] p. 368. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
enpw1	⊢ (A ≈ B ↔ ℘1A ≈ ℘1B)

Proof of Theorem enpw1

Dummy variables a b f g w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. 2 ⊢ (A ≈ B → (A ∈ V ∧ B ∈ V))
2		brex 4690	. . 3 ⊢ (℘1A ≈ ℘1B → (℘1A ∈ V ∧ ℘1B ∈ V))
3		pw1exb 4327	. . . 4 ⊢ (℘1A ∈ V ↔ A ∈ V)
4		pw1exb 4327	. . . 4 ⊢ (℘1B ∈ V ↔ B ∈ V)
5	3, 4	anbi12i 678	. . 3 ⊢ ((℘1A ∈ V ∧ ℘1B ∈ V) ↔ (A ∈ V ∧ B ∈ V))
6	2, 5	sylib 188	. 2 ⊢ (℘1A ≈ ℘1B → (A ∈ V ∧ B ∈ V))
7		breq1 4643	. . . 4 ⊢ (a = A → (a ≈ b ↔ A ≈ b))
8		pw1eq 4144	. . . . 5 ⊢ (a = A → ℘1a = ℘1A)
9	8	breq1d 4650	. . . 4 ⊢ (a = A → (℘1a ≈ ℘1b ↔ ℘1A ≈ ℘1b))
10	7, 9	bibi12d 312	. . 3 ⊢ (a = A → ((a ≈ b ↔ ℘1a ≈ ℘1b) ↔ (A ≈ b ↔ ℘1A ≈ ℘1b)))
11		breq2 4644	. . . 4 ⊢ (b = B → (A ≈ b ↔ A ≈ B))
12		pw1eq 4144	. . . . 5 ⊢ (b = B → ℘1b = ℘1B)
13	12	breq2d 4652	. . . 4 ⊢ (b = B → (℘1A ≈ ℘1b ↔ ℘1A ≈ ℘1B))
14	11, 13	bibi12d 312	. . 3 ⊢ (b = B → ((A ≈ b ↔ ℘1A ≈ ℘1b) ↔ (A ≈ B ↔ ℘1A ≈ ℘1B)))
15		bren 6031	. . . . 5 ⊢ (a ≈ b ↔ ∃f f:a–1-1-onto→b)
16		f1ofun 5290	. . . . . . . . . 10 ⊢ (f:a–1-1-onto→b → Fun f)
17		funsi 5521	. . . . . . . . . 10 ⊢ (Fun f → Fun SI f)
18	16, 17	syl 15	. . . . . . . . 9 ⊢ (f:a–1-1-onto→b → Fun SI f)
19		f1odm 5291	. . . . . . . . . 10 ⊢ (f:a–1-1-onto→b → dom f = a)
20		dmsi 5520	. . . . . . . . . . 11 ⊢ dom SI f = ℘1dom f
21		pw1eq 4144	. . . . . . . . . . 11 ⊢ (dom f = a → ℘1dom f = ℘1a)
22	20, 21	syl5eq 2397	. . . . . . . . . 10 ⊢ (dom f = a → dom SI f = ℘1a)
23	19, 22	syl 15	. . . . . . . . 9 ⊢ (f:a–1-1-onto→b → dom SI f = ℘1a)
24		df-fn 4791	. . . . . . . . 9 ⊢ ( SI f Fn ℘1a ↔ (Fun SI f ∧ dom SI f = ℘1a))
25	18, 23, 24	sylanbrc 645	. . . . . . . 8 ⊢ (f:a–1-1-onto→b → SI f Fn ℘1a)
26		f1of1 5287	. . . . . . . . . . 11 ⊢ (f:a–1-1-onto→b → f:a–1-1→b)
27		df-f1 4793	. . . . . . . . . . . 12 ⊢ (f:a–1-1→b ↔ (f:a–→b ∧ Fun ◡f))
28	27	simprbi 450	. . . . . . . . . . 11 ⊢ (f:a–1-1→b → Fun ◡f)
29		funsi 5521	. . . . . . . . . . 11 ⊢ (Fun ◡f → Fun SI ◡f)
30	26, 28, 29	3syl 18	. . . . . . . . . 10 ⊢ (f:a–1-1-onto→b → Fun SI ◡f)
31		cnvsi 5519	. . . . . . . . . . 11 ⊢ ◡ SI f = SI ◡f
32	31	funeqi 5129	. . . . . . . . . 10 ⊢ (Fun ◡ SI f ↔ Fun SI ◡f)
33	30, 32	sylibr 203	. . . . . . . . 9 ⊢ (f:a–1-1-onto→b → Fun ◡ SI f)
34		f1ofo 5294	. . . . . . . . . 10 ⊢ (f:a–1-1-onto→b → f:a–onto→b)
35		forn 5273	. . . . . . . . . 10 ⊢ (f:a–onto→b → ran f = b)
36		rnsi 5522	. . . . . . . . . . . 12 ⊢ ran SI f = ℘1ran f
37		dfrn4 4905	. . . . . . . . . . . 12 ⊢ ran SI f = dom ◡ SI f
38	36, 37	eqtr3i 2375	. . . . . . . . . . 11 ⊢ ℘1ran f = dom ◡ SI f
39		pw1eq 4144	. . . . . . . . . . 11 ⊢ (ran f = b → ℘1ran f = ℘1b)
40	38, 39	syl5eqr 2399	. . . . . . . . . 10 ⊢ (ran f = b → dom ◡ SI f = ℘1b)
41	34, 35, 40	3syl 18	. . . . . . . . 9 ⊢ (f:a–1-1-onto→b → dom ◡ SI f = ℘1b)
42		df-fn 4791	. . . . . . . . 9 ⊢ (◡ SI f Fn ℘1b ↔ (Fun ◡ SI f ∧ dom ◡ SI f = ℘1b))
43	33, 41, 42	sylanbrc 645	. . . . . . . 8 ⊢ (f:a–1-1-onto→b → ◡ SI f Fn ℘1b)
44		dff1o4 5295	. . . . . . . 8 ⊢ ( SI f:℘1a–1-1-onto→℘1b ↔ ( SI f Fn ℘1a ∧ ◡ SI f Fn ℘1b))
45	25, 43, 44	sylanbrc 645	. . . . . . 7 ⊢ (f:a–1-1-onto→b → SI f:℘1a–1-1-onto→℘1b)
46		vex 2863	. . . . . . . . 9 ⊢ f ∈ V
47	46	siex 4754	. . . . . . . 8 ⊢ SI f ∈ V
48	47	f1oen 6034	. . . . . . 7 ⊢ ( SI f:℘1a–1-1-onto→℘1b → ℘1a ≈ ℘1b)
49	45, 48	syl 15	. . . . . 6 ⊢ (f:a–1-1-onto→b → ℘1a ≈ ℘1b)
50	49	exlimiv 1634	. . . . 5 ⊢ (∃f f:a–1-1-onto→b → ℘1a ≈ ℘1b)
51	15, 50	sylbi 187	. . . 4 ⊢ (a ≈ b → ℘1a ≈ ℘1b)
52		bren 6031	. . . . 5 ⊢ (℘1a ≈ ℘1b ↔ ∃g g:℘1a–1-1-onto→℘1b)
53		f1ofun 5290	. . . . . . . . . . . . . 14 ⊢ (g:℘1a–1-1-onto→℘1b → Fun g)
54		fununiq 5518	. . . . . . . . . . . . . . . 16 ⊢ ((Fun g ∧ {x}g{y} ∧ {x}g{z}) → {y} = {z})
55		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ y ∈ V
56	55	sneqb 3877	. . . . . . . . . . . . . . . 16 ⊢ ({y} = {z} ↔ y = z)
57	54, 56	sylib 188	. . . . . . . . . . . . . . 15 ⊢ ((Fun g ∧ {x}g{y} ∧ {x}g{z}) → y = z)
58	57	3expib 1154	. . . . . . . . . . . . . 14 ⊢ (Fun g → (({x}g{y} ∧ {x}g{z}) → y = z))
59	53, 58	syl 15	. . . . . . . . . . . . 13 ⊢ (g:℘1a–1-1-onto→℘1b → (({x}g{y} ∧ {x}g{z}) → y = z))
60	59	alrimivv 1632	. . . . . . . . . . . 12 ⊢ (g:℘1a–1-1-onto→℘1b → ∀y∀z(({x}g{y} ∧ {x}g{z}) → y = z))
61		sneq 3745	. . . . . . . . . . . . . 14 ⊢ (y = z → {y} = {z})
62	61	breq2d 4652	. . . . . . . . . . . . 13 ⊢ (y = z → ({x}g{y} ↔ {x}g{z}))
63	62	mo4 2237	. . . . . . . . . . . 12 ⊢ (∃*y{x}g{y} ↔ ∀y∀z(({x}g{y} ∧ {x}g{z}) → y = z))
64	60, 63	sylibr 203	. . . . . . . . . . 11 ⊢ (g:℘1a–1-1-onto→℘1b → ∃*y{x}g{y})
65	64	alrimiv 1631	. . . . . . . . . 10 ⊢ (g:℘1a–1-1-onto→℘1b → ∀x∃*y{x}g{y})
66		funopab 5140	. . . . . . . . . 10 ⊢ (Fun {⟨x, y⟩ ∣ {x}g{y}} ↔ ∀x∃*y{x}g{y})
67	65, 66	sylibr 203	. . . . . . . . 9 ⊢ (g:℘1a–1-1-onto→℘1b → Fun {⟨x, y⟩ ∣ {x}g{y}})
68		dmopab 4916	. . . . . . . . . 10 ⊢ dom {⟨x, y⟩ ∣ {x}g{y}} = {x ∣ ∃y{x}g{y}}
69		eldm 4899	. . . . . . . . . . . . . . 15 ⊢ ({x} ∈ dom g ↔ ∃z{x}gz)
70		brelrn 4961	. . . . . . . . . . . . . . . . 17 ⊢ ({x}gz → z ∈ ran g)
71		f1ofo 5294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (g:℘1a–1-1-onto→℘1b → g:℘1a–onto→℘1b)
72		forn 5273	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (g:℘1a–onto→℘1b → ran g = ℘1b)
73	71, 72	syl 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ (g:℘1a–1-1-onto→℘1b → ran g = ℘1b)
74	73	eleq2d 2420	. . . . . . . . . . . . . . . . . . 19 ⊢ (g:℘1a–1-1-onto→℘1b → (z ∈ ran g ↔ z ∈ ℘1b))
75		elpw1 4145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (z ∈ ℘1b ↔ ∃w ∈ b z = {w})
76		breq2 4644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (z = {w} → ({x}gz ↔ {x}g{w}))
77		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ w ∈ V
78		sneq 3745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (y = w → {y} = {w})
79	78	breq2d 4652	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (y = w → ({x}g{y} ↔ {x}g{w}))
80	77, 79	spcev 2947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({x}g{w} → ∃y{x}g{y})
81	76, 80	syl6bi 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (z = {w} → ({x}gz → ∃y{x}g{y}))
82	81	rexlimivw 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃w ∈ b z = {w} → ({x}gz → ∃y{x}g{y}))
83	75, 82	sylbi 187	. . . . . . . . . . . . . . . . . . 19 ⊢ (z ∈ ℘1b → ({x}gz → ∃y{x}g{y}))
84	74, 83	syl6bi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (g:℘1a–1-1-onto→℘1b → (z ∈ ran g → ({x}gz → ∃y{x}g{y})))
85	84	com23 72	. . . . . . . . . . . . . . . . 17 ⊢ (g:℘1a–1-1-onto→℘1b → ({x}gz → (z ∈ ran g → ∃y{x}g{y})))
86	70, 85	mpdi 38	. . . . . . . . . . . . . . . 16 ⊢ (g:℘1a–1-1-onto→℘1b → ({x}gz → ∃y{x}g{y}))
87	86	exlimdv 1636	. . . . . . . . . . . . . . 15 ⊢ (g:℘1a–1-1-onto→℘1b → (∃z{x}gz → ∃y{x}g{y}))
88	69, 87	syl5bi 208	. . . . . . . . . . . . . 14 ⊢ (g:℘1a–1-1-onto→℘1b → ({x} ∈ dom g → ∃y{x}g{y}))
89		breldm 4912	. . . . . . . . . . . . . . 15 ⊢ ({x}g{y} → {x} ∈ dom g)
90	89	exlimiv 1634	. . . . . . . . . . . . . 14 ⊢ (∃y{x}g{y} → {x} ∈ dom g)
91	88, 90	impbid1 194	. . . . . . . . . . . . 13 ⊢ (g:℘1a–1-1-onto→℘1b → ({x} ∈ dom g ↔ ∃y{x}g{y}))
92		f1odm 5291	. . . . . . . . . . . . . 14 ⊢ (g:℘1a–1-1-onto→℘1b → dom g = ℘1a)
93	92	eleq2d 2420	. . . . . . . . . . . . 13 ⊢ (g:℘1a–1-1-onto→℘1b → ({x} ∈ dom g ↔ {x} ∈ ℘1a))
94	91, 93	bitr3d 246	. . . . . . . . . . . 12 ⊢ (g:℘1a–1-1-onto→℘1b → (∃y{x}g{y} ↔ {x} ∈ ℘1a))
95		snelpw1 4147	. . . . . . . . . . . 12 ⊢ ({x} ∈ ℘1a ↔ x ∈ a)
96	94, 95	syl6bb 252	. . . . . . . . . . 11 ⊢ (g:℘1a–1-1-onto→℘1b → (∃y{x}g{y} ↔ x ∈ a))
97	96	eqabcdv 2470	. . . . . . . . . 10 ⊢ (g:℘1a–1-1-onto→℘1b → {x ∣ ∃y{x}g{y}} = a)
98	68, 97	syl5eq 2397	. . . . . . . . 9 ⊢ (g:℘1a–1-1-onto→℘1b → dom {⟨x, y⟩ ∣ {x}g{y}} = a)
99		df-fn 4791	. . . . . . . . 9 ⊢ ({⟨x, y⟩ ∣ {x}g{y}} Fn a ↔ (Fun {⟨x, y⟩ ∣ {x}g{y}} ∧ dom {⟨x, y⟩ ∣ {x}g{y}} = a))
100	67, 98, 99	sylanbrc 645	. . . . . . . 8 ⊢ (g:℘1a–1-1-onto→℘1b → {⟨x, y⟩ ∣ {x}g{y}} Fn a)
101		f1ocnv 5300	. . . . . . . . . . . . . . 15 ⊢ (g:℘1a–1-1-onto→℘1b → ◡g:℘1b–1-1-onto→℘1a)
102		f1ofun 5290	. . . . . . . . . . . . . . 15 ⊢ (◡g:℘1b–1-1-onto→℘1a → Fun ◡g)
103		fununiq 5518	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun ◡g ∧ {y}◡g{x} ∧ {y}◡g{z}) → {x} = {z})
104	103	3expib 1154	. . . . . . . . . . . . . . . 16 ⊢ (Fun ◡g → (({y}◡g{x} ∧ {y}◡g{z}) → {x} = {z}))
105		brcnv 4893	. . . . . . . . . . . . . . . . 17 ⊢ ({y}◡g{x} ↔ {x}g{y})
106		brcnv 4893	. . . . . . . . . . . . . . . . 17 ⊢ ({y}◡g{z} ↔ {z}g{y})
107	105, 106	anbi12i 678	. . . . . . . . . . . . . . . 16 ⊢ (({y}◡g{x} ∧ {y}◡g{z}) ↔ ({x}g{y} ∧ {z}g{y}))
108		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ x ∈ V
109	108	sneqb 3877	. . . . . . . . . . . . . . . 16 ⊢ ({x} = {z} ↔ x = z)
110	104, 107, 109	3imtr3g 260	. . . . . . . . . . . . . . 15 ⊢ (Fun ◡g → (({x}g{y} ∧ {z}g{y}) → x = z))
111	101, 102, 110	3syl 18	. . . . . . . . . . . . . 14 ⊢ (g:℘1a–1-1-onto→℘1b → (({x}g{y} ∧ {z}g{y}) → x = z))
112	111	alrimivv 1632	. . . . . . . . . . . . 13 ⊢ (g:℘1a–1-1-onto→℘1b → ∀x∀z(({x}g{y} ∧ {z}g{y}) → x = z))
113		sneq 3745	. . . . . . . . . . . . . . 15 ⊢ (x = z → {x} = {z})
114	113	breq1d 4650	. . . . . . . . . . . . . 14 ⊢ (x = z → ({x}g{y} ↔ {z}g{y}))
115	114	mo4 2237	. . . . . . . . . . . . 13 ⊢ (∃*x{x}g{y} ↔ ∀x∀z(({x}g{y} ∧ {z}g{y}) → x = z))
116	112, 115	sylibr 203	. . . . . . . . . . . 12 ⊢ (g:℘1a–1-1-onto→℘1b → ∃*x{x}g{y})
117	116	alrimiv 1631	. . . . . . . . . . 11 ⊢ (g:℘1a–1-1-onto→℘1b → ∀y∃*x{x}g{y})
118		funopab 5140	. . . . . . . . . . 11 ⊢ (Fun {⟨y, x⟩ ∣ {x}g{y}} ↔ ∀y∃*x{x}g{y})
119	117, 118	sylibr 203	. . . . . . . . . 10 ⊢ (g:℘1a–1-1-onto→℘1b → Fun {⟨y, x⟩ ∣ {x}g{y}})
120		dmopab 4916	. . . . . . . . . . 11 ⊢ dom {⟨y, x⟩ ∣ {x}g{y}} = {y ∣ ∃x{x}g{y}}
121		elrn 4897	. . . . . . . . . . . . . . . 16 ⊢ ({y} ∈ ran g ↔ ∃z zg{y})
122		breldm 4912	. . . . . . . . . . . . . . . . . 18 ⊢ (zg{y} → z ∈ dom g)
123	92	eleq2d 2420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (g:℘1a–1-1-onto→℘1b → (z ∈ dom g ↔ z ∈ ℘1a))
124		elpw1 4145	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (z ∈ ℘1a ↔ ∃w ∈ a z = {w})
125		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (z = {w} → (zg{y} ↔ {w}g{y}))
126		sneq 3745	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (x = w → {x} = {w})
127	126	breq1d 4650	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (x = w → ({x}g{y} ↔ {w}g{y}))
128	77, 127	spcev 2947	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({w}g{y} → ∃x{x}g{y})
129	125, 128	syl6bi 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (z = {w} → (zg{y} → ∃x{x}g{y}))
130	129	rexlimivw 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃w ∈ a z = {w} → (zg{y} → ∃x{x}g{y}))
131	124, 130	sylbi 187	. . . . . . . . . . . . . . . . . . . 20 ⊢ (z ∈ ℘1a → (zg{y} → ∃x{x}g{y}))
132	123, 131	syl6bi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (g:℘1a–1-1-onto→℘1b → (z ∈ dom g → (zg{y} → ∃x{x}g{y})))
133	132	com23 72	. . . . . . . . . . . . . . . . . 18 ⊢ (g:℘1a–1-1-onto→℘1b → (zg{y} → (z ∈ dom g → ∃x{x}g{y})))
134	122, 133	mpdi 38	. . . . . . . . . . . . . . . . 17 ⊢ (g:℘1a–1-1-onto→℘1b → (zg{y} → ∃x{x}g{y}))
135	134	exlimdv 1636	. . . . . . . . . . . . . . . 16 ⊢ (g:℘1a–1-1-onto→℘1b → (∃z zg{y} → ∃x{x}g{y}))
136	121, 135	syl5bi 208	. . . . . . . . . . . . . . 15 ⊢ (g:℘1a–1-1-onto→℘1b → ({y} ∈ ran g → ∃x{x}g{y}))
137		brelrn 4961	. . . . . . . . . . . . . . . 16 ⊢ ({x}g{y} → {y} ∈ ran g)
138	137	exlimiv 1634	. . . . . . . . . . . . . . 15 ⊢ (∃x{x}g{y} → {y} ∈ ran g)
139	136, 138	impbid1 194	. . . . . . . . . . . . . 14 ⊢ (g:℘1a–1-1-onto→℘1b → ({y} ∈ ran g ↔ ∃x{x}g{y}))
140	73	eleq2d 2420	. . . . . . . . . . . . . 14 ⊢ (g:℘1a–1-1-onto→℘1b → ({y} ∈ ran g ↔ {y} ∈ ℘1b))
141	139, 140	bitr3d 246	. . . . . . . . . . . . 13 ⊢ (g:℘1a–1-1-onto→℘1b → (∃x{x}g{y} ↔ {y} ∈ ℘1b))
142		snelpw1 4147	. . . . . . . . . . . . 13 ⊢ ({y} ∈ ℘1b ↔ y ∈ b)
143	141, 142	syl6bb 252	. . . . . . . . . . . 12 ⊢ (g:℘1a–1-1-onto→℘1b → (∃x{x}g{y} ↔ y ∈ b))
144	143	eqabcdv 2470	. . . . . . . . . . 11 ⊢ (g:℘1a–1-1-onto→℘1b → {y ∣ ∃x{x}g{y}} = b)
145	120, 144	syl5eq 2397	. . . . . . . . . 10 ⊢ (g:℘1a–1-1-onto→℘1b → dom {⟨y, x⟩ ∣ {x}g{y}} = b)
146		df-fn 4791	. . . . . . . . . 10 ⊢ ({⟨y, x⟩ ∣ {x}g{y}} Fn b ↔ (Fun {⟨y, x⟩ ∣ {x}g{y}} ∧ dom {⟨y, x⟩ ∣ {x}g{y}} = b))
147	119, 145, 146	sylanbrc 645	. . . . . . . . 9 ⊢ (g:℘1a–1-1-onto→℘1b → {⟨y, x⟩ ∣ {x}g{y}} Fn b)
148		cnvopab 5031	. . . . . . . . . 10 ⊢ ◡{⟨x, y⟩ ∣ {x}g{y}} = {⟨y, x⟩ ∣ {x}g{y}}
149	148	fneq1i 5179	. . . . . . . . 9 ⊢ (◡{⟨x, y⟩ ∣ {x}g{y}} Fn b ↔ {⟨y, x⟩ ∣ {x}g{y}} Fn b)
150	147, 149	sylibr 203	. . . . . . . 8 ⊢ (g:℘1a–1-1-onto→℘1b → ◡{⟨x, y⟩ ∣ {x}g{y}} Fn b)
151		dff1o4 5295	. . . . . . . 8 ⊢ ({⟨x, y⟩ ∣ {x}g{y}}:a–1-1-onto→b ↔ ({⟨x, y⟩ ∣ {x}g{y}} Fn a ∧ ◡{⟨x, y⟩ ∣ {x}g{y}} Fn b))
152	100, 150, 151	sylanbrc 645	. . . . . . 7 ⊢ (g:℘1a–1-1-onto→℘1b → {⟨x, y⟩ ∣ {x}g{y}}:a–1-1-onto→b)
153		enpw1lem1 6062	. . . . . . . 8 ⊢ {⟨x, y⟩ ∣ {x}g{y}} ∈ V
154	153	f1oen 6034	. . . . . . 7 ⊢ ({⟨x, y⟩ ∣ {x}g{y}}:a–1-1-onto→b → a ≈ b)
155	152, 154	syl 15	. . . . . 6 ⊢ (g:℘1a–1-1-onto→℘1b → a ≈ b)
156	155	exlimiv 1634	. . . . 5 ⊢ (∃g g:℘1a–1-1-onto→℘1b → a ≈ b)
157	52, 156	sylbi 187	. . . 4 ⊢ (℘1a ≈ ℘1b → a ≈ b)
158	51, 157	impbii 180	. . 3 ⊢ (a ≈ b ↔ ℘1a ≈ ℘1b)
159	10, 14, 158	vtocl2g 2919	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A ≈ B ↔ ℘1A ≈ ℘1B))
160	1, 6, 159	pm5.21nii 342	1 ⊢ (A ≈ B ↔ ℘1A ≈ ℘1B)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  {cab 2339  ∃wrex 2616  Vcvv 2860  {csn 3738  ℘₁cpw1 4136  {copab 4623   class class class wbr 4640   SI csi 4721  ^◡ccnv 4772  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777  –→wf 4778  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781   ≈ cen 6029

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 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 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 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-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-en 6030

This theorem is referenced by:  ncpw1  6153  ce0addcnnul  6180  cenc  6182  tc11  6229