Theorem ceex 6175

Description: Cardinal exponentiation is stratified. (Contributed by SF, 3-Mar-2015.)

Assertion

Ref	Expression
ceex	⊢ ↑c ∈ V

Proof of Theorem ceex

Dummy variables a b g m n x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ce 6107	. . 3 ⊢ ↑c = (n ∈ NC , m ∈ NC ↦ {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
2		snex 4112	. . . . . . . . . 10 ⊢ {x} ∈ V
3	2	otelins2 5792	. . . . . . . . 9 ⊢ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ Ins2 Ins3 ( S ∘ SI Pw1Fn ) ↔ ⟨{{a}}, ⟨n, m⟩⟩ ∈ Ins3 ( S ∘ SI Pw1Fn ))
4		vex 2863	. . . . . . . . . 10 ⊢ m ∈ V
5	4	otelins3 5793	. . . . . . . . 9 ⊢ (⟨{{a}}, ⟨n, m⟩⟩ ∈ Ins3 ( S ∘ SI Pw1Fn ) ↔ ⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ))
6		ceexlem1 6174	. . . . . . . . 9 ⊢ (⟨{{a}}, n⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘1a ∈ n)
7	3, 5, 6	3bitri 262	. . . . . . . 8 ⊢ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ Ins2 Ins3 ( S ∘ SI Pw1Fn ) ↔ ℘1a ∈ n)
8		elimapw11c 4949	. . . . . . . . 9 ⊢ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c) ↔ ∃b⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ ( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )))
9		elin 3220	. . . . . . . . . . 11 ⊢ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ ( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∧ ⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )))
10		snex 4112	. . . . . . . . . . . . . 14 ⊢ {{a}} ∈ V
11	10	otelins2 5792	. . . . . . . . . . . . 13 ⊢ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ↔ ⟨{{b}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ Ins2 Ins2 ( S ∘ SI Pw1Fn ))
12	2	otelins2 5792	. . . . . . . . . . . . 13 ⊢ (⟨{{b}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ Ins2 Ins2 ( S ∘ SI Pw1Fn ) ↔ ⟨{{b}}, ⟨n, m⟩⟩ ∈ Ins2 ( S ∘ SI Pw1Fn ))
13		vex 2863	. . . . . . . . . . . . . . 15 ⊢ n ∈ V
14	13	otelins2 5792	. . . . . . . . . . . . . 14 ⊢ (⟨{{b}}, ⟨n, m⟩⟩ ∈ Ins2 ( S ∘ SI Pw1Fn ) ↔ ⟨{{b}}, m⟩ ∈ ( S ∘ SI Pw1Fn ))
15		ceexlem1 6174	. . . . . . . . . . . . . 14 ⊢ (⟨{{b}}, m⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘1b ∈ m)
16	14, 15	bitri 240	. . . . . . . . . . . . 13 ⊢ (⟨{{b}}, ⟨n, m⟩⟩ ∈ Ins2 ( S ∘ SI Pw1Fn ) ↔ ℘1b ∈ m)
17	11, 12, 16	3bitri 262	. . . . . . . . . . . 12 ⊢ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ↔ ℘1b ∈ m)
18	13, 4	opex 4589	. . . . . . . . . . . . . 14 ⊢ ⟨n, m⟩ ∈ V
19	18	oqelins4 5795	. . . . . . . . . . . . 13 ⊢ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ ⟨{{b}}, ⟨{{a}}, {x}⟩⟩ ∈ SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
20		snex 4112	. . . . . . . . . . . . . 14 ⊢ {b} ∈ V
21		snex 4112	. . . . . . . . . . . . . 14 ⊢ {a} ∈ V
22		vex 2863	. . . . . . . . . . . . . 14 ⊢ x ∈ V
23	20, 21, 22	otsnelsi3 5806	. . . . . . . . . . . . 13 ⊢ (⟨{{b}}, ⟨{{a}}, {x}⟩⟩ ∈ SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ ⟨{b}, ⟨{a}, x⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
24		elrn2 4898	. . . . . . . . . . . . . 14 ⊢ (⟨{b}, ⟨{a}, x⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ ∃g⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ))
25		elin 3220	. . . . . . . . . . . . . . . 16 ⊢ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∧ ⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins2 Ins2 ≈ ))
26	22	oqelins4 5795	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ ⟨g, ⟨{b}, {a}⟩⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c))
27	20, 21	opex 4589	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨{b}, {a}⟩ ∈ V
28		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ f ∈ V
29	28	brfns 5834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (f Fns b ↔ f Fn b)
30		brco 4884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (f( S ∘ Image2nd )a ↔ ∃x(fImage2nd x ∧ x S a))
31	28, 22	brimage 5794	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (fImage2nd x ↔ x = (2nd “ f))
32		dfrn5 5509	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ran f = (2nd “ f)
33	32	eqeq2i 2363	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (x = ran f ↔ x = (2nd “ f))
34	31, 33	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (fImage2nd x ↔ x = ran f)
35		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ a ∈ V
36	22, 35	brsset 4759	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (x S a ↔ x ⊆ a)
37	34, 36	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((fImage2nd x ∧ x S a) ↔ (x = ran f ∧ x ⊆ a))
38	37	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃x(fImage2nd x ∧ x S a) ↔ ∃x(x = ran f ∧ x ⊆ a))
39	28	rnex 5108	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ran f ∈ V
40		sseq1 3293	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (x = ran f → (x ⊆ a ↔ ran f ⊆ a))
41	39, 40	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃x(x = ran f ∧ x ⊆ a) ↔ ran f ⊆ a)
42	30, 38, 41	3bitri 262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (f( S ∘ Image2nd )a ↔ ran f ⊆ a)
43	29, 42	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((f Fns b ∧ f( S ∘ Image2nd )a) ↔ (f Fn b ∧ ran f ⊆ a))
44		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (f( Fns ⊗ ( S ∘ Image2nd ))⟨b, a⟩ ↔ ⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )))
45		trtxp 5782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (f( Fns ⊗ ( S ∘ Image2nd ))⟨b, a⟩ ↔ (f Fns b ∧ f( S ∘ Image2nd )a))
46	44, 45	bitr3i 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )) ↔ (f Fns b ∧ f( S ∘ Image2nd )a))
47		df-f 4792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (f:b–→a ↔ (f Fn b ∧ ran f ⊆ a))
48	43, 46, 47	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )) ↔ f:b–→a)
49		vex 2863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ b ∈ V
50	28, 49, 35	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{f}, ⟨{b}, {a}⟩⟩ ∈ SI3 ( Fns ⊗ ( S ∘ Image2nd )) ↔ ⟨f, ⟨b, a⟩⟩ ∈ ( Fns ⊗ ( S ∘ Image2nd )))
51	35, 49, 28	elmap 6018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (f ∈ (a ↑m b) ↔ f:b–→a)
52	48, 50, 51	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{f}, ⟨{b}, {a}⟩⟩ ∈ SI3 ( Fns ⊗ ( S ∘ Image2nd )) ↔ f ∈ (a ↑m b))
53	27, 52	releqel 5808	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨g, ⟨{b}, {a}⟩⟩ ∈ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ g = (a ↑m b))
54	26, 53	bitri 240	. . . . . . . . . . . . . . . . 17 ⊢ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ↔ g = (a ↑m b))
55	20	otelins2 5792	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins2 Ins2 ≈ ↔ ⟨g, ⟨{a}, x⟩⟩ ∈ Ins2 ≈ )
56	21	otelins2 5792	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨g, ⟨{a}, x⟩⟩ ∈ Ins2 ≈ ↔ ⟨g, x⟩ ∈ ≈ )
57		df-br 4641	. . . . . . . . . . . . . . . . . . 19 ⊢ (g ≈ x ↔ ⟨g, x⟩ ∈ ≈ )
58	56, 57	bitr4i 243	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨g, ⟨{a}, x⟩⟩ ∈ Ins2 ≈ ↔ g ≈ x)
59		ensym 6038	. . . . . . . . . . . . . . . . . 18 ⊢ (g ≈ x ↔ x ≈ g)
60	55, 58, 59	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins2 Ins2 ≈ ↔ x ≈ g)
61	54, 60	anbi12i 678	. . . . . . . . . . . . . . . 16 ⊢ ((⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∧ ⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ Ins2 Ins2 ≈ ) ↔ (g = (a ↑m b) ∧ x ≈ g))
62	25, 61	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ (g = (a ↑m b) ∧ x ≈ g))
63	62	exbii 1582	. . . . . . . . . . . . . 14 ⊢ (∃g⟨g, ⟨{b}, ⟨{a}, x⟩⟩⟩ ∈ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ ∃g(g = (a ↑m b) ∧ x ≈ g))
64		ovex 5552	. . . . . . . . . . . . . . 15 ⊢ (a ↑m b) ∈ V
65		breq2 4644	. . . . . . . . . . . . . . 15 ⊢ (g = (a ↑m b) → (x ≈ g ↔ x ≈ (a ↑m b)))
66	64, 65	ceqsexv 2895	. . . . . . . . . . . . . 14 ⊢ (∃g(g = (a ↑m b) ∧ x ≈ g) ↔ x ≈ (a ↑m b))
67	24, 63, 66	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟨{b}, ⟨{a}, x⟩⟩ ∈ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ x ≈ (a ↑m b))
68	19, 23, 67	3bitri 262	. . . . . . . . . . . 12 ⊢ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ↔ x ≈ (a ↑m b))
69	17, 68	anbi12i 678	. . . . . . . . . . 11 ⊢ ((⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∧ ⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ (℘1b ∈ m ∧ x ≈ (a ↑m b)))
70	9, 69	bitri 240	. . . . . . . . . 10 ⊢ (⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ ( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ (℘1b ∈ m ∧ x ≈ (a ↑m b)))
71	70	exbii 1582	. . . . . . . . 9 ⊢ (∃b⟨{{b}}, ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩⟩ ∈ ( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ↔ ∃b(℘1b ∈ m ∧ x ≈ (a ↑m b)))
72	8, 71	bitri 240	. . . . . . . 8 ⊢ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c) ↔ ∃b(℘1b ∈ m ∧ x ≈ (a ↑m b)))
73	7, 72	anbi12i 678	. . . . . . 7 ⊢ ((⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∧ ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) ↔ (℘1a ∈ n ∧ ∃b(℘1b ∈ m ∧ x ≈ (a ↑m b))))
74		elin 3220	. . . . . . 7 ⊢ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ ( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) ↔ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∧ ⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)))
75		3anass 938	. . . . . . . . 9 ⊢ ((℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b)) ↔ (℘1a ∈ n ∧ (℘1b ∈ m ∧ x ≈ (a ↑m b))))
76	75	exbii 1582	. . . . . . . 8 ⊢ (∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b)) ↔ ∃b(℘1a ∈ n ∧ (℘1b ∈ m ∧ x ≈ (a ↑m b))))
77		19.42v 1905	. . . . . . . 8 ⊢ (∃b(℘1a ∈ n ∧ (℘1b ∈ m ∧ x ≈ (a ↑m b))) ↔ (℘1a ∈ n ∧ ∃b(℘1b ∈ m ∧ x ≈ (a ↑m b))))
78	76, 77	bitri 240	. . . . . . 7 ⊢ (∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b)) ↔ (℘1a ∈ n ∧ ∃b(℘1b ∈ m ∧ x ≈ (a ↑m b))))
79	73, 74, 78	3bitr4i 268	. . . . . 6 ⊢ (⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ ( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) ↔ ∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b)))
80	79	exbii 1582	. . . . 5 ⊢ (∃a⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ ( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) ↔ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b)))
81		elimapw11c 4949	. . . . 5 ⊢ (⟨{x}, ⟨n, m⟩⟩ ∈ (( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) “ ℘11c) ↔ ∃a⟨{{a}}, ⟨{x}, ⟨n, m⟩⟩⟩ ∈ ( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)))
82		breq1 4643	. . . . . . . 8 ⊢ (g = x → (g ≈ (a ↑m b) ↔ x ≈ (a ↑m b)))
83	82	3anbi3d 1258	. . . . . . 7 ⊢ (g = x → ((℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b)) ↔ (℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b))))
84	83	2exbidv 1628	. . . . . 6 ⊢ (g = x → (∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b)) ↔ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b))))
85	22, 84	elab 2986	. . . . 5 ⊢ (x ∈ {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))} ↔ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ x ≈ (a ↑m b)))
86	80, 81, 85	3bitr4i 268	. . . 4 ⊢ (⟨{x}, ⟨n, m⟩⟩ ∈ (( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) “ ℘11c) ↔ x ∈ {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
87	86	releqmpt2 5810	. . 3 ⊢ ((( NC × NC ) × V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) “ ℘11c)) “ 1c)) = (n ∈ NC , m ∈ NC ↦ {g ∣ ∃a∃b(℘1a ∈ n ∧ ℘1b ∈ m ∧ g ≈ (a ↑m b))})
88	1, 87	eqtr4i 2376	. 2 ⊢ ↑c = ((( NC × NC ) × V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) “ ℘11c)) “ 1c))
89		ncsex 6112	. . 3 ⊢ NC ∈ V
90		ssetex 4745	. . . . . . . 8 ⊢ S ∈ V
91		pw1fnex 5853	. . . . . . . . 9 ⊢ Pw1Fn ∈ V
92	91	siex 4754	. . . . . . . 8 ⊢ SI Pw1Fn ∈ V
93	90, 92	coex 4751	. . . . . . 7 ⊢ ( S ∘ SI Pw1Fn ) ∈ V
94	93	ins3ex 5799	. . . . . 6 ⊢ Ins3 ( S ∘ SI Pw1Fn ) ∈ V
95	94	ins2ex 5798	. . . . 5 ⊢ Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∈ V
96	93	ins2ex 5798	. . . . . . . . 9 ⊢ Ins2 ( S ∘ SI Pw1Fn ) ∈ V
97	96	ins2ex 5798	. . . . . . . 8 ⊢ Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∈ V
98	97	ins2ex 5798	. . . . . . 7 ⊢ Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∈ V
99	90	ins3ex 5799	. . . . . . . . . . . . . . 15 ⊢ Ins3 S ∈ V
100		fnsex 5833	. . . . . . . . . . . . . . . . . 18 ⊢ Fns ∈ V
101		2ndex 5113	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2nd ∈ V
102	101	imageex 5802	. . . . . . . . . . . . . . . . . . 19 ⊢ Image2nd ∈ V
103	90, 102	coex 4751	. . . . . . . . . . . . . . . . . 18 ⊢ ( S ∘ Image2nd ) ∈ V
104	100, 103	txpex 5786	. . . . . . . . . . . . . . . . 17 ⊢ ( Fns ⊗ ( S ∘ Image2nd )) ∈ V
105	104	si3ex 5807	. . . . . . . . . . . . . . . 16 ⊢ SI3 ( Fns ⊗ ( S ∘ Image2nd )) ∈ V
106	105	ins2ex 5798	. . . . . . . . . . . . . . 15 ⊢ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd )) ∈ V
107	99, 106	symdifex 4109	. . . . . . . . . . . . . 14 ⊢ ( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) ∈ V
108		1cex 4143	. . . . . . . . . . . . . 14 ⊢ 1c ∈ V
109	107, 108	imaex 4748	. . . . . . . . . . . . 13 ⊢ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∈ V
110	109	complex 4105	. . . . . . . . . . . 12 ⊢ ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∈ V
111	110	ins4ex 5800	. . . . . . . . . . 11 ⊢ Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∈ V
112		enex 6032	. . . . . . . . . . . . 13 ⊢ ≈ ∈ V
113	112	ins2ex 5798	. . . . . . . . . . . 12 ⊢ Ins2 ≈ ∈ V
114	113	ins2ex 5798	. . . . . . . . . . 11 ⊢ Ins2 Ins2 ≈ ∈ V
115	111, 114	inex 4106	. . . . . . . . . 10 ⊢ ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ∈ V
116	115	rnex 5108	. . . . . . . . 9 ⊢ ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ∈ V
117	116	si3ex 5807	. . . . . . . 8 ⊢ SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ∈ V
118	117	ins4ex 5800	. . . . . . 7 ⊢ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ ) ∈ V
119	98, 118	inex 4106	. . . . . 6 ⊢ ( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) ∈ V
120	108	pw1ex 4304	. . . . . 6 ⊢ ℘11c ∈ V
121	119, 120	imaex 4748	. . . . 5 ⊢ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c) ∈ V
122	95, 121	inex 4106	. . . 4 ⊢ ( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) ∈ V
123	122, 120	imaex 4748	. . 3 ⊢ (( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) “ ℘11c) ∈ V
124	89, 89, 123	mpt2exlem 5812	. 2 ⊢ ((( NC × NC ) × V) ∖ (( Ins2 S ⊕ Ins3 (( Ins2 Ins3 ( S ∘ SI Pw1Fn ) ∩ (( Ins2 Ins2 Ins2 ( S ∘ SI Pw1Fn ) ∩ Ins4 SI3 ran ( Ins4 ∼ (( Ins3 S ⊕ Ins2 SI3 ( Fns ⊗ ( S ∘ Image2nd ))) “ 1c) ∩ Ins2 Ins2 ≈ )) “ ℘11c)) “ ℘11c)) “ 1c)) ∈ V
125	88, 124	eqeltri 2423	1 ⊢ ↑c ∈ V

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209   ⊕ csymdif 3210   ⊆ wss 3258  {csn 3738  1_cc1c 4135  ℘₁cpw1 4136  ⟨cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   ∘ ccom 4722   “ cima 4723   × cxp 4771  ran crn 4774   Fn wfn 4777  –→wf 4778  2^nd c2nd 4784  (class class class)co 5526   ↦ cmpt2 5654   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752  Imagecimage 5754   Ins4 cins4 5756   SI₃ csi3 5758   Fns cfns 5762   Pw1Fn cpw1fn 5766   ↑_m cmap 6000   ≈ cen 6029   NC cncs 6089   ↑_c cce 6097

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-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-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-ce 6107

This theorem is referenced by:  ce0nn  6181  spacvallem1  6282