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Theorem swapex 4742

Description: The Swap function is a set. (Contributed by SF, 6-Jan-2015.)

**Assertion**
Ref	Expression
swapex	⊢ Swap ∈ V

**Proof of Theorem swapex**
Step	Hyp	Ref	Expression
1		dfswap2 4741	. 2 ⊢ Swap = (( ∼ (( Ins2_k Ins2_k S_k ⊕ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) “_k ℘₁1_c) “_k V)
2		ssetkex 4294	. . . . . . . . 9 ⊢ S_k ∈ V
3	2	ins2kex 4307	. . . . . . . 8 ⊢ Ins2_k S_k ∈ V
4	3	ins2kex 4307	. . . . . . 7 ⊢ Ins2_k Ins2_k S_k ∈ V
5		addcexlem 4382	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) ∈ V
6		1cex 4142	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1_c ∈ V
7	6	pw1ex 4303	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℘₁1_c ∈ V
8	7	pw1ex 4303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℘₁℘₁1_c ∈ V
9	5, 8	imakex 4300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
10	9	imagekex 4312	. . . . . . . . . . . . . . . . . . . 20 ⊢ Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
11		nncex 4396	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Nn ∈ V
12		vvex 4109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ V ∈ V
13	11, 12	xpkex 4289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( Nn ×_k V) ∈ V
14	10, 13	inex 4105	. . . . . . . . . . . . . . . . . . 19 ⊢ (Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∈ V
15		idkex 4314	. . . . . . . . . . . . . . . . . . . 20 ⊢ I_k ∈ V
16	11	complex 4104	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∼ Nn ∈ V
17	16, 12	xpkex 4289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( ∼ Nn ×_k V) ∈ V
18	15, 17	inex 4105	. . . . . . . . . . . . . . . . . . 19 ⊢ ( I_k ∩ ( ∼ Nn ×_k V)) ∈ V
19	14, 18	unex 4106	. . . . . . . . . . . . . . . . . 18 ⊢ ((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
20	19	imagekex 4312	. . . . . . . . . . . . . . . . 17 ⊢ Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
21	20	cnvkex 4287	. . . . . . . . . . . . . . . 16 ⊢ ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
22	21	sikex 4297	. . . . . . . . . . . . . . 15 ⊢ SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
23	2, 22	cokex 4310	. . . . . . . . . . . . . 14 ⊢ ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
24	23	ins3kex 4308	. . . . . . . . . . . . 13 ⊢ Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
25	24	ins2kex 4307	. . . . . . . . . . . 12 ⊢ Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
26	21, 2	cokex 4310	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∈ V
27		snex 4111	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {{0_c}} ∈ V
28	27, 12	xpkex 4289	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({{0_c}} ×_k V) ∈ V
29	26, 28	unex 4106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V)) ∈ V
30	29	ins3kex 4308	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V)) ∈ V
31	3, 30	symdifex 4108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) ∈ V
32	31, 8	imakex 4300	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
33	32	complex 4104	. . . . . . . . . . . . . . . . . . 19 ⊢ ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
34	33	sikex 4297	. . . . . . . . . . . . . . . . . 18 ⊢ SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
35	34	ins3kex 4308	. . . . . . . . . . . . . . . . 17 ⊢ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
36	3, 35	inex 4105	. . . . . . . . . . . . . . . 16 ⊢ ( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) ∈ V
37	36, 8	imakex 4300	. . . . . . . . . . . . . . 15 ⊢ (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
38	37	sikex 4297	. . . . . . . . . . . . . 14 ⊢ SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
39	38	sikex 4297	. . . . . . . . . . . . 13 ⊢ SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
40	39	ins3kex 4308	. . . . . . . . . . . 12 ⊢ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
41	25, 40	unex 4106	. . . . . . . . . . 11 ⊢ ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∈ V
42	41	ins2kex 4307	. . . . . . . . . 10 ⊢ Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∈ V
43	20	sikex 4297	. . . . . . . . . . . . . . 15 ⊢ SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
44	43	sikex 4297	. . . . . . . . . . . . . 14 ⊢ SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
45	44	sikex 4297	. . . . . . . . . . . . 13 ⊢ SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
46	45	sikex 4297	. . . . . . . . . . . 12 ⊢ SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
47	46	sikex 4297	. . . . . . . . . . 11 ⊢ SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
48	47	ins3kex 4308	. . . . . . . . . 10 ⊢ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
49	42, 48	inex 4105	. . . . . . . . 9 ⊢ ( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
50	8	pw1ex 4303	. . . . . . . . . . . 12 ⊢ ℘₁℘₁℘₁1_c ∈ V
51	50	pw1ex 4303	. . . . . . . . . . 11 ⊢ ℘₁℘₁℘₁℘₁1_c ∈ V
52	51	pw1ex 4303	. . . . . . . . . 10 ⊢ ℘₁℘₁℘₁℘₁℘₁1_c ∈ V
53	52	pw1ex 4303	. . . . . . . . 9 ⊢ ℘₁℘₁℘₁℘₁℘₁℘₁1_c ∈ V
54	49, 53	imakex 4300	. . . . . . . 8 ⊢ (( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∈ V
55	23	sikex 4297	. . . . . . . . . . . . . 14 ⊢ SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
56	55	sikex 4297	. . . . . . . . . . . . 13 ⊢ SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
57	56	ins3kex 4308	. . . . . . . . . . . 12 ⊢ Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∈ V
58	37	ins3kex 4308	. . . . . . . . . . . . 13 ⊢ Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
59	58	ins2kex 4307	. . . . . . . . . . . 12 ⊢ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
60	57, 59	unex 4106	. . . . . . . . . . 11 ⊢ ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∈ V
61	60	ins2kex 4307	. . . . . . . . . 10 ⊢ Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∈ V
62	34	sikex 4297	. . . . . . . . . . . . . 14 ⊢ SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
63	62	sikex 4297	. . . . . . . . . . . . 13 ⊢ SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
64	63	sikex 4297	. . . . . . . . . . . 12 ⊢ SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
65	64	sikex 4297	. . . . . . . . . . 11 ⊢ SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
66	65	ins3kex 4308	. . . . . . . . . 10 ⊢ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c) ∈ V
67	61, 66	inex 4105	. . . . . . . . 9 ⊢ ( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) ∈ V
68	67, 53	imakex 4300	. . . . . . . 8 ⊢ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∈ V
69	54, 68	unex 4106	. . . . . . 7 ⊢ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c)) ∈ V
70	4, 69	symdifex 4108	. . . . . 6 ⊢ ( Ins2_k Ins2_k S_k ⊕ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c))) ∈ V
71	70, 51	imakex 4300	. . . . 5 ⊢ (( Ins2_k Ins2_k S_k ⊕ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) ∈ V
72	71	complex 4104	. . . 4 ⊢ ∼ (( Ins2_k Ins2_k S_k ⊕ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) ∈ V
73	72, 7	imakex 4300	. . 3 ⊢ ( ∼ (( Ins2_k Ins2_k S_k ⊕ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) “_k ℘₁1_c) ∈ V
74	73, 12	imakex 4300	. 2 ⊢ (( ∼ (( Ins2_k Ins2_k S_k ⊕ ((( Ins2_k ( Ins2_k Ins3_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins3_k SI_k SI_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k Image_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c) ∪ (( Ins2_k ( Ins3_k SI_k SI_k ( S_k ∘_k SI_k ^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V)))) ∪ Ins2_k Ins3_k (( Ins2_k S_k ∩ Ins3_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁1_c)) ∩ Ins3_k SI_k SI_k SI_k SI_k SI_k ∼ (( Ins2_k S_k ⊕ Ins3_k ((^◡_kImage_k((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k ℘₁℘₁℘₁℘₁1_c)) “_k ℘₁℘₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∘_k S_k ) ∪ ({{0_c}} ×_k V))) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁℘₁℘₁℘₁1_c))) “_k ℘₁℘₁℘₁℘₁1_c) “_k ℘₁1_c) “_k V) ∈ V
75	1, 74	eqeltri 2423	1 ⊢ Swap ∈ V

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 ⊕ csymdif 3209 {csn 3737 1_cc1c 4134 ℘₁cpw1 4135 ×_k cxpk 4174 ^◡_kccnvk 4175 Ins2_k cins2k 4176 Ins3_k cins3k 4177 “_k cimak 4179 ∘_k ccomk 4180 SI_k csik 4181 Image_kcimagek 4182 S_k cssetk 4183 I_k cidk 4184 Nn cnnc 4373 0_cc0c 4374 Swap cswap 4718

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-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-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 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-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-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-opab 4623 df-swap 4724

This theorem is referenced by: cnvexg 5101 2ndex 5112 composeex 5820 domfnex 5870 connexex 5913 symex 5916 enex 6031 cnven 6045 xpcomen 6052

W3C validator