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Theorem tfinrelkex 4487

Description: The expression at the core of eqtfinrelk 4486 exists. (Contributed by SF, 30-Jan-2015.)

**Assertion**
Ref	Expression
tfinrelkex	⊢ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∖ ({{∅}} ×_k V))) ∈ V

**Proof of Theorem tfinrelkex**
Step	Hyp	Ref	Expression
1		snex 4111	. . 3 ⊢ {{∅}} ∈ V
2		snex 4111	. . 3 ⊢ {∅} ∈ V
3	1, 2	xpkex 4289	. 2 ⊢ ({{∅}} ×_k {∅}) ∈ V
4		ssetkex 4294	. . . . . . 7 ⊢ S_k ∈ V
5	4	ins2kex 4307	. . . . . 6 ⊢ Ins2_k S_k ∈ V
6	4	cnvkex 4287	. . . . . . . . . 10 ⊢ ^◡_k S_k ∈ V
7	6	ins3kex 4308	. . . . . . . . 9 ⊢ Ins3_k ^◡_k S_k ∈ V
8		nncex 4396	. . . . . . . . . . . . . . 15 ⊢ Nn ∈ V
9		vvex 4109	. . . . . . . . . . . . . . 15 ⊢ V ∈ V
10	8, 9	xpkex 4289	. . . . . . . . . . . . . 14 ⊢ ( Nn ×_k V) ∈ V
11	4	sikex 4297	. . . . . . . . . . . . . . . . 17 ⊢ SI_k S_k ∈ V
12	11	ins2kex 4307	. . . . . . . . . . . . . . . 16 ⊢ Ins2_k SI_k S_k ∈ V
13		1cex 4142	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1_c ∈ V
14	13	pwex 4329	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℘1_c ∈ V
15	14, 9	xpkex 4289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (℘1_c ×_k V) ∈ V
16	4	ins3kex 4308	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ins3_k S_k ∈ V
17	16, 12	symdifex 4108	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) ∈ V
18	13	pw1ex 4303	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ℘₁1_c ∈ V
19	18	pw1ex 4303	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℘₁℘₁1_c ∈ V
20	19	pw1ex 4303	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℘₁℘₁℘₁1_c ∈ V
21	17, 20	imakex 4300	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c) ∈ V
22	15, 21	difex 4107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
23	22	sikex 4297	. . . . . . . . . . . . . . . . . . . 20 ⊢ SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
24	23	ins3kex 4308	. . . . . . . . . . . . . . . . . . 19 ⊢ Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∈ V
25	24, 5	inex 4105	. . . . . . . . . . . . . . . . . 18 ⊢ ( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) ∈ V
26	25, 19	imakex 4300	. . . . . . . . . . . . . . . . 17 ⊢ (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∈ V
27	26	ins3kex 4308	. . . . . . . . . . . . . . . 16 ⊢ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c) ∈ V
28	12, 27	inex 4105	. . . . . . . . . . . . . . 15 ⊢ ( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) ∈ V
29	28, 20	imakex 4300	. . . . . . . . . . . . . 14 ⊢ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c) ∈ V
30	10, 29	inex 4105	. . . . . . . . . . . . 13 ⊢ (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ∈ V
31	30	ins2kex 4307	. . . . . . . . . . . 12 ⊢ Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ∈ V
32		idkex 4314	. . . . . . . . . . . . 13 ⊢ I_k ∈ V
33	32	ins3kex 4308	. . . . . . . . . . . 12 ⊢ Ins3_k I_k ∈ V
34	31, 33	symdifex 4108	. . . . . . . . . . 11 ⊢ ( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) ∈ V
35	34, 18	imakex 4300	. . . . . . . . . 10 ⊢ (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c) ∈ V
36	35	ins2kex 4307	. . . . . . . . 9 ⊢ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c) ∈ V
37	7, 36	difex 4107	. . . . . . . 8 ⊢ ( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) ∈ V
38	37, 18	imakex 4300	. . . . . . 7 ⊢ (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c) ∈ V
39	38	ins3kex 4308	. . . . . 6 ⊢ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c) ∈ V
40	5, 39	symdifex 4108	. . . . 5 ⊢ ( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) ∈ V
41	40, 19	imakex 4300	. . . 4 ⊢ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
42	41	complex 4104	. . 3 ⊢ ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∈ V
43	1, 9	xpkex 4289	. . 3 ⊢ ({{∅}} ×_k V) ∈ V
44	42, 43	difex 4107	. 2 ⊢ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∖ ({{∅}} ×_k V)) ∈ V
45	3, 44	unex 4106	1 ⊢ (({{∅}} ×_k {∅}) ∪ ( ∼ (( Ins2_k S_k ⊕ Ins3_k (( Ins3_k ^◡_k S_k ∖ Ins2_k (( Ins2_k (( Nn ×_k V) ∩ (( Ins2_k SI_k S_k ∩ Ins3_k (( Ins3_k SI_k ((℘1_c ×_k V) ∖ (( Ins3_k S_k ⊕ Ins2_k SI_k S_k ) “_k ℘₁℘₁℘₁1_c)) ∩ Ins2_k S_k ) “_k ℘₁℘₁1_c)) “_k ℘₁℘₁℘₁1_c)) ⊕ Ins3_k I_k ) “_k ℘₁1_c)) “_k ℘₁1_c)) “_k ℘₁℘₁1_c) ∖ ({{∅}} ×_k V))) ∈ V

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 ⊕ csymdif 3209 ∅c0 3550 ℘cpw 3722 {csn 3737 1_cc1c 4134 ℘₁cpw1 4135 ×_k cxpk 4174 ^◡_kccnvk 4175 Ins2_k cins2k 4176 Ins3_k cins3k 4177 “_k cimak 4179 SI_k csik 4181 S_k cssetk 4183 I_k cidk 4184 Nn cnnc 4373

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

This theorem is referenced by: sfintfinlem1 4531 tfinnnlem1 4533 vfinspss 4551 vfinspclt 4552 vfinncsp 4554

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