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Theorem phiexg 4572

Description: The phi operator preserves sethood. (Contributed by SF, 3-Feb-2015.)

**Assertion**
Ref	Expression
phiexg	⊢ (A ∈ V → Phi A ∈ V)

**Proof of Theorem phiexg**
Step	Hyp	Ref	Expression
1		dfphi2 4570	. 2 ⊢ Phi A = (((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k A)
2		addcexlem 4383	. . . . . . 7 ⊢ ( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) ∈ V
3		1cex 4143	. . . . . . . . 9 ⊢ 1_c ∈ V
4	3	pw1ex 4304	. . . . . . . 8 ⊢ 𝒫₁1_c ∈ V
5	4	pw1ex 4304	. . . . . . 7 ⊢ 𝒫₁𝒫₁1_c ∈ V
6	2, 5	imakex 4301	. . . . . 6 ⊢ (( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∈ V
7	6	imagekex 4313	. . . . 5 ⊢ Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∈ V
8		nncex 4397	. . . . . 6 ⊢ Nn ∈ V
9		vvex 4110	. . . . . 6 ⊢ V ∈ V
10	8, 9	xpkex 4290	. . . . 5 ⊢ ( Nn ×_k V) ∈ V
11	7, 10	inex 4106	. . . 4 ⊢ (Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∈ V
12		idkex 4315	. . . . 5 ⊢ I_k ∈ V
13	8	complex 4105	. . . . . 6 ⊢ ∼ Nn ∈ V
14	13, 9	xpkex 4290	. . . . 5 ⊢ ( ∼ Nn ×_k V) ∈ V
15	12, 14	inex 4106	. . . 4 ⊢ ( I_k ∩ ( ∼ Nn ×_k V)) ∈ V
16	11, 15	unex 4107	. . 3 ⊢ ((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V
17		imakexg 4300	. . 3 ⊢ ((((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) ∈ V ∧ A ∈ V) → (((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k A) ∈ V)
18	16, 17	mpan 651	. 2 ⊢ (A ∈ V → (((Image_k(( Ins3_k ∼ (( Ins3_k S_k ∩ Ins2_k S_k ) “_k 𝒫₁𝒫₁1_c) ∖ (( Ins2_k Ins2_k S_k ⊕ ( Ins2_k Ins3_k S_k ∪ Ins3_k SI_k SI_k S_k )) “_k 𝒫₁𝒫₁𝒫₁𝒫₁1_c)) “_k 𝒫₁𝒫₁1_c) ∩ ( Nn ×_k V)) ∪ ( I_k ∩ ( ∼ Nn ×_k V))) “_k A) ∈ V)
19	1, 18	syl5eqel 2437	1 ⊢ (A ∈ V → Phi A ∈ V)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 1710 Vcvv 2860 ∼ ccompl 3206 ∖ cdif 3207 ∪ cun 3208 ∩ cin 3209 ⊕ csymdif 3210 1_cc1c 4135 𝒫₁cpw1 4136 ×_k cxpk 4175 Ins2_k cins2k 4177 Ins3_k cins3k 4178 “_k cimak 4180 SI_k csik 4182 Image_kcimagek 4183 S_k cssetk 4184 I_k cidk 4185 Nn cnnc 4374 Phi cphi 4563

This proof depends on 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 proof 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

This theorem is used by: phiex 4573

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