df1st2 - New Foundations Explorer

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Theorem df1st2 4738

Description: Express the

function via the set construction functions. (Contributed by SF, 4-Feb-2015.)

**Assertion**
Ref	Expression
df1st2	⋃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 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c

Proof of Theorem df1st2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opkex 4113	. . . . . 6
2	1	elimak 4259	. . . . 5 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
3		df-rex 2620	. . . . . 6 ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
4		elpw11c 4147	. . . . . . . . . 10 ₁ 1_c
5	4	anbi1i 676	. . . . . . . . 9 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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		19.41v 1901	. . . . . . . . 9 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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	5, 6	bitr4i 243	. . . . . . . 8 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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	7	exbii 1582	. . . . . . 7 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
9		excom 1741	. . . . . . 7 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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	8, 9	bitr4i 243	. . . . . 6 ₁ 1_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
11	3, 10	bitri 240	. . . . 5 ₁ 1_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
12	2, 11	bitri 240	. . . 4 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
13		snex 4111	. . . . . . 7
14		opkeq1 4059	. . . . . . . 8
15	14	eleq1d 2419	. . . . . . 7 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
16	13, 15	ceqsexv 2894	. . . . . 6 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
17		vex 2862	. . . . . . 7
18		vex 2862	. . . . . . 7
19		vex 2862	. . . . . . 7
20	17, 18, 19	setconslem7 4737	. . . . . 6 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
21	16, 20	bitri 240	. . . . 5 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
22	21	exbii 1582	. . . 4 $/\$ ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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
23	12, 22	bitri 240	. . 3 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c
24	23	opabbii 4626	. 2 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c
25		setconslem4 4734	. 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 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c
26		df-1st 4723	. 2
27	24, 25, 26	3eqtr4ri 2384	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 ∼ Ins2_k Ins3_k S_k Ins2_k Ins2_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 Ins3_k SI_k SI_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_c

Colors of variables: wff setvar class

Syntax hints: $/\$ wa 358

wex 1541

wceq 1642

wcel 1710

wrex 2615

cvv 2859 ∼ ccompl 3205

cdif 3206

cun 3207

cin 3208

csymdif 3209

csn 3737

copk 4057 ⋃₁cuni1 4133 1_cc1c 4134

₁ cpw1 4135

_k cxpk 4174

_kccnvk 4175 Ins2_k cins2k 4176 Ins3_k cins3k 4177

_kcimak 4179

_k ccomk 4180 SI_k csik 4181 Image_kcimagek 4182 S_k cssetk 4183

_k cidk 4184 Nn cnnc 4373 0_cc0c 4374

cop 4561

copab 4622

c1st 4717

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-1st 4723

This theorem is referenced by: 1stex 4739

W3C validator