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Theorem coexg 4750

Description: The composition of two sets is a set. (Contributed by SF, 7-Jan-2015.)

**Assertion**
Ref	Expression
coexg	$/\$

**Proof of Theorem coexg**
Step	Hyp	Ref	Expression
1		dfco1 4749	. 2 ⋃₁⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
2		vvex 4110	. . . . . . 7
3	2, 2	xpkex 4290	. . . . . 6 _k
4	3, 2	xpkex 4290	. . . . 5 _k _k
5		setconslem5 4736	. . . . . 6 ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
6	5	cnvkex 4288	. . . . 5 _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
7	4, 6	inex 4106	. . . 4 _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
8	2, 3	xpkex 4290	. . . . . . 7 _k _k
9	8, 5	inex 4106	. . . . . 6 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c
10		pw1exg 4303	. . . . . . 7 ₁
11		pw1exg 4303	. . . . . . 7 ₁ ₁ ₁
12	10, 11	syl 15	. . . . . 6 ₁ ₁
13		imakexg 4300	. . . . . 6 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ₁ ₁ _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
14	9, 12, 13	sylancr 644	. . . . 5 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
15		pw1exg 4303	. . . . . . 7 ₁
16		pw1exg 4303	. . . . . . 7 ₁ ₁ ₁
17	15, 16	syl 15	. . . . . 6 ₁ ₁
18		imakexg 4300	. . . . . 6 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ ₁ ₁ _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
19	9, 17, 18	sylancr 644	. . . . 5 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
20		cokexg 4310	. . . . 5 _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ $/\$ _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
21	14, 19, 20	syl2an 463	. . . 4 $/\$ _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
22		imakexg 4300	. . . 4 _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c $/\$ _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
23	7, 21, 22	sylancr 644	. . 3 $/\$ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
24		uni1exg 4293	. . 3 _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ ⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
25		uni1exg 4293	. . 3 ⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ ⋃₁⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
26	23, 24, 25	3syl 18	. 2 $/\$ ⋃₁⋃₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁ _k _k _k ∼ Ins3_k SI_k SI_k S_k Ins2_k Ins3_k S_k _k SI_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k Ins2_k Ins2_k S_k Ins3_k SI_k ∼ Ins2_k S_k Ins3_k _kImage_kImage_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c Nn _k _k ∼ Nn _k _k S_k 0_c _k _k₁ ₁ 1_c_k₁ ₁ 1_c_k₁ ₁ ₁ ₁ 1_c_k₁ ₁
27	1, 26	syl5eqel 2437	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 358

wcel 1710

cvv 2860 ∼ ccompl 3206

cdif 3207

cun 3208

cin 3209

csymdif 3210

csn 3738 ⋃₁cuni1 4134 1_cc1c 4135

₁ cpw1 4136

_k cxpk 4175

_kccnvk 4176 Ins2_k cins2k 4177 Ins3_k cins3k 4178

_kcimak 4180

_k ccomk 4181 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184

_k cidk 4185 Nn cnnc 4374 0_cc0c 4375

ccom 4722

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 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-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 2479 df-ne 2519 df-ral 2620 df-rex 2621 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-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-addc 4379 df-nnc 4380 df-phi 4566 df-op 4567 df-opab 4624 df-br 4641 df-co 4727

This theorem is referenced by: coex 4751 txpexg 5785 imageexg 5801 composevalg 5818 pprodexg 5838 fullfunexg 5860 enmap2lem3 6066 enmap1lem3 6072

W3C validator