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Theorem ssetex 4744

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

**Assertion**
Ref	Expression
ssetex	⊢ S ∈ V

**Proof of Theorem ssetex**
Step	Hyp	Ref	Expression
1		dfsset2 4743	. 2 ⊢ S = ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k S_k )
2		vvex 4109	. . . . . . . 8 ⊢ V ∈ V
3	2, 2	xpkex 4289	. . . . . . 7 ⊢ (V ×_k V) ∈ V
4	3, 2	xpkex 4289	. . . . . 6 ⊢ ((V ×_k V) ×_k V) ∈ V
5		setconslem5 4735	. . . . . . 7 ⊢ ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c) ∈ V
6	5	cnvkex 4287	. . . . . 6 ⊢ ^◡_k ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c) ∈ V
7	4, 6	inex 4105	. . . . 5 ⊢ (((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c)) ∈ V
8		ssetkex 4294	. . . . 5 ⊢ S_k ∈ V
9	7, 8	imakex 4300	. . . 4 ⊢ ((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k S_k ) ∈ V
10	9	uni1ex 4293	. . 3 ⊢ ⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k S_k ) ∈ V
11	10	uni1ex 4293	. 2 ⊢ ⋃₁⋃₁((((V ×_k V) ×_k V) ∩ ^◡_k ∼ (( Ins3_k SI_k SI_k S_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)))) ∪ Ins2_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))) “_k ℘₁℘₁℘₁℘₁1_c)) “_k S_k ) ∈ V
12	1, 11	eqeltri 2423	1 ⊢ S ∈ 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 ⋃₁cuni1 4133 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 S csset 4719

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-sset 4725

This theorem is referenced by: idex 5504 imageexg 5800 mptexlem 5810 mpt2exlem 5811 cupex 5816 composeex 5820 disjex 5823 addcfnex 5824 funsex 5828 crossex 5850 pw1fnex 5852 domfnex 5870 ranfnex 5871 clos1ex 5876 transex 5910 refex 5911 antisymex 5912 connexex 5913 foundex 5914 extex 5915 symex 5916 ssetpov 5944 qsexg 5982 enprmaplem1 6076 nenpw1pwlem1 6084 lecex 6115 mucex 6133 ovcelem1 6171 ceex 6174 tcfnex 6244 nmembers1lem1 6268 nchoicelem11 6299 nchoicelem16 6304 nchoicelem18 6306

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