Theorem tcfnex 6245

Description: The stratified T raising function is a set. (Contributed by SF, 18-Mar-2015.)

Assertion

Ref	Expression
tcfnex	⊢ TcFn ∈ V

Proof of Theorem tcfnex

Dummy variables q p x z y t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tcfn 6108	. . 3 ⊢ TcFn = (x ∈ 1c ↦ Tc ∪x)
2		oteltxp 5783	. . . . . . 7 ⊢ (⟨z, ⟨{y}, x⟩⟩ ∈ (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ↔ (⟨z, {y}⟩ ∈ ◡ S ∧ ⟨z, x⟩ ∈ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )))
3		df-br 4641	. . . . . . . . 9 ⊢ (z◡ S {y} ↔ ⟨z, {y}⟩ ∈ ◡ S )
4		brcnv 4893	. . . . . . . . . 10 ⊢ (z◡ S {y} ↔ {y} S z)
5		vex 2863	. . . . . . . . . . 11 ⊢ y ∈ V
6		vex 2863	. . . . . . . . . . 11 ⊢ z ∈ V
7	5, 6	brssetsn 4760	. . . . . . . . . 10 ⊢ ({y} S z ↔ y ∈ z)
8	4, 7	bitri 240	. . . . . . . . 9 ⊢ (z◡ S {y} ↔ y ∈ z)
9	3, 8	bitr3i 242	. . . . . . . 8 ⊢ (⟨z, {y}⟩ ∈ ◡ S ↔ y ∈ z)
10		vex 2863	. . . . . . . . . . 11 ⊢ x ∈ V
11	6, 10	opex 4589	. . . . . . . . . 10 ⊢ ⟨z, x⟩ ∈ V
12	11	elcompl 3226	. . . . . . . . 9 ⊢ (⟨z, x⟩ ∈ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ¬ ⟨z, x⟩ ∈ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ))
13		elrn2 4898	. . . . . . . . . . 11 ⊢ (⟨z, x⟩ ∈ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ∃p⟨p, ⟨z, x⟩⟩ ∈ ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ))
14		elsymdif 3224	. . . . . . . . . . . . 13 ⊢ (⟨p, ⟨z, x⟩⟩ ∈ ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ¬ (⟨p, ⟨z, x⟩⟩ ∈ Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ ⟨p, ⟨z, x⟩⟩ ∈ Ins3 I ))
15	6	otelins2 5792	. . . . . . . . . . . . . . 15 ⊢ (⟨p, ⟨z, x⟩⟩ ∈ Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ ⟨p, x⟩ ∈ (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)))
16		elin 3220	. . . . . . . . . . . . . . 15 ⊢ (⟨p, x⟩ ∈ (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ (⟨p, x⟩ ∈ ( NC × V) ∧ ⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)))
17		opelxp 4812	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨p, x⟩ ∈ ( NC × V) ↔ (p ∈ NC ∧ x ∈ V))
18	10, 17	mpbiran2 885	. . . . . . . . . . . . . . . . 17 ⊢ (⟨p, x⟩ ∈ ( NC × V) ↔ p ∈ NC )
19	18	anbi1i 676	. . . . . . . . . . . . . . . 16 ⊢ ((⟨p, x⟩ ∈ ( NC × V) ∧ ⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ (p ∈ NC ∧ ⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)))
20		ncseqnc 6129	. . . . . . . . . . . . . . . . . . 19 ⊢ (p ∈ NC → (p = Nc ℘1q ↔ ℘1q ∈ p))
21	20	rexbidv 2636	. . . . . . . . . . . . . . . . . 18 ⊢ (p ∈ NC → (∃q ∈ ∪xp = Nc ℘1q ↔ ∃q ∈ ∪x℘1q ∈ p))
22		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{{q}}, ⟨p, x⟩⟩ ∈ (( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (⟨{{q}}, p⟩ ∈ ( S ∘ SI Pw1Fn ) ∧ ⟨{{q}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c)))
23		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {q} ∈ V
24	23	brsnsi1 5776	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({{q}} SI Pw1Fn u ↔ ∃t(u = {t} ∧ {q} Pw1Fn t))
25	24	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({{q}} SI Pw1Fn u ∧ u S p) ↔ (∃t(u = {t} ∧ {q} Pw1Fn t) ∧ u S p))
26		19.41v 1901	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃t((u = {t} ∧ {q} Pw1Fn t) ∧ u S p) ↔ (∃t(u = {t} ∧ {q} Pw1Fn t) ∧ u S p))
27	25, 26	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({{q}} SI Pw1Fn u ∧ u S p) ↔ ∃t((u = {t} ∧ {q} Pw1Fn t) ∧ u S p))
28	27	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃u({{q}} SI Pw1Fn u ∧ u S p) ↔ ∃u∃t((u = {t} ∧ {q} Pw1Fn t) ∧ u S p))
29		excom 1741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃u∃t((u = {t} ∧ {q} Pw1Fn t) ∧ u S p) ↔ ∃t∃u((u = {t} ∧ {q} Pw1Fn t) ∧ u S p))
30		anass 630	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((u = {t} ∧ {q} Pw1Fn t) ∧ u S p) ↔ (u = {t} ∧ ({q} Pw1Fn t ∧ u S p)))
31	30	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃u((u = {t} ∧ {q} Pw1Fn t) ∧ u S p) ↔ ∃u(u = {t} ∧ ({q} Pw1Fn t ∧ u S p)))
32		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {t} ∈ V
33		breq1 4643	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (u = {t} → (u S p ↔ {t} S p))
34	33	anbi2d 684	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (u = {t} → (({q} Pw1Fn t ∧ u S p) ↔ ({q} Pw1Fn t ∧ {t} S p)))
35	32, 34	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃u(u = {t} ∧ ({q} Pw1Fn t ∧ u S p)) ↔ ({q} Pw1Fn t ∧ {t} S p))
36		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ q ∈ V
37	36	brpw1fn 5855	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({q} Pw1Fn t ↔ t = ℘1q)
38		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ t ∈ V
39		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ p ∈ V
40	38, 39	brssetsn 4760	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({t} S p ↔ t ∈ p)
41	37, 40	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({q} Pw1Fn t ∧ {t} S p) ↔ (t = ℘1q ∧ t ∈ p))
42	31, 35, 41	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃u((u = {t} ∧ {q} Pw1Fn t) ∧ u S p) ↔ (t = ℘1q ∧ t ∈ p))
43	42	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃t∃u((u = {t} ∧ {q} Pw1Fn t) ∧ u S p) ↔ ∃t(t = ℘1q ∧ t ∈ p))
44	28, 29, 43	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃u({{q}} SI Pw1Fn u ∧ u S p) ↔ ∃t(t = ℘1q ∧ t ∈ p))
45		opelco 4885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{{q}}, p⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ∃u({{q}} SI Pw1Fn u ∧ u S p))
46		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (℘1q ∈ p ↔ ∃t(t = ℘1q ∧ t ∈ p))
47	44, 45, 46	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{{q}}, p⟩ ∈ ( S ∘ SI Pw1Fn ) ↔ ℘1q ∈ p)
48		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨{t}, ⟨{{q}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ) ↔ (⟨{t}, {{q}}⟩ ∈ SI ◡ S ∧ ⟨{t}, x⟩ ∈ S ))
49		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({t} SI ◡ S {{q}} ↔ ⟨{t}, {{q}}⟩ ∈ SI ◡ S )
50	38, 23	brsnsi 5774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({t} SI ◡ S {{q}} ↔ t◡ S {q})
51		brcnv 4893	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (t◡ S {q} ↔ {q} S t)
52	36, 38	brssetsn 4760	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ({q} S t ↔ q ∈ t)
53	50, 51, 52	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({t} SI ◡ S {{q}} ↔ q ∈ t)
54	49, 53	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨{t}, {{q}}⟩ ∈ SI ◡ S ↔ q ∈ t)
55	38, 10	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨{t}, x⟩ ∈ S ↔ t ∈ x)
56	54, 55	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((⟨{t}, {{q}}⟩ ∈ SI ◡ S ∧ ⟨{t}, x⟩ ∈ S ) ↔ (q ∈ t ∧ t ∈ x))
57	48, 56	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨{t}, ⟨{{q}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ) ↔ (q ∈ t ∧ t ∈ x))
58	57	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃t⟨{t}, ⟨{{q}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ) ↔ ∃t(q ∈ t ∧ t ∈ x))
59		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨{{q}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c) ↔ ∃t⟨{t}, ⟨{{q}}, x⟩⟩ ∈ ( SI ◡ S ⊗ S ))
60		eluni 3895	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (q ∈ ∪x ↔ ∃t(q ∈ t ∧ t ∈ x))
61	58, 59, 60	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{{q}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c) ↔ q ∈ ∪x)
62	47, 61	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⟨{{q}}, p⟩ ∈ ( S ∘ SI Pw1Fn ) ∧ ⟨{{q}}, x⟩ ∈ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (℘1q ∈ p ∧ q ∈ ∪x))
63		ancom 437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((℘1q ∈ p ∧ q ∈ ∪x) ↔ (q ∈ ∪x ∧ ℘1q ∈ p))
64	22, 62, 63	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{{q}}, ⟨p, x⟩⟩ ∈ (( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ (q ∈ ∪x ∧ ℘1q ∈ p))
65	64	exbii 1582	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃q⟨{{q}}, ⟨p, x⟩⟩ ∈ (( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ↔ ∃q(q ∈ ∪x ∧ ℘1q ∈ p))
66		elimapw11c 4949	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ↔ ∃q⟨{{q}}, ⟨p, x⟩⟩ ∈ (( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)))
67		df-rex 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃q ∈ ∪x℘1q ∈ p ↔ ∃q(q ∈ ∪x ∧ ℘1q ∈ p))
68	65, 66, 67	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ↔ ∃q ∈ ∪x℘1q ∈ p)
69	21, 68	syl6rbbr 255	. . . . . . . . . . . . . . . . 17 ⊢ (p ∈ NC → (⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ↔ ∃q ∈ ∪xp = Nc ℘1q))
70	69	pm5.32i 618	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ NC ∧ ⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ (p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q))
71	19, 70	bitri 240	. . . . . . . . . . . . . . 15 ⊢ ((⟨p, x⟩ ∈ ( NC × V) ∧ ⟨p, x⟩ ∈ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ (p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q))
72	15, 16, 71	3bitri 262	. . . . . . . . . . . . . 14 ⊢ (⟨p, ⟨z, x⟩⟩ ∈ Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ (p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q))
73	10	otelins3 5793	. . . . . . . . . . . . . . 15 ⊢ (⟨p, ⟨z, x⟩⟩ ∈ Ins3 I ↔ ⟨p, z⟩ ∈ I )
74		df-br 4641	. . . . . . . . . . . . . . . 16 ⊢ (p I z ↔ ⟨p, z⟩ ∈ I )
75	6	ideq 4871	. . . . . . . . . . . . . . . 16 ⊢ (p I z ↔ p = z)
76	74, 75	bitr3i 242	. . . . . . . . . . . . . . 15 ⊢ (⟨p, z⟩ ∈ I ↔ p = z)
77	73, 76	bitri 240	. . . . . . . . . . . . . 14 ⊢ (⟨p, ⟨z, x⟩⟩ ∈ Ins3 I ↔ p = z)
78	72, 77	bibi12i 306	. . . . . . . . . . . . 13 ⊢ ((⟨p, ⟨z, x⟩⟩ ∈ Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ↔ ⟨p, ⟨z, x⟩⟩ ∈ Ins3 I ) ↔ ((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z))
79	14, 78	xchbinx 301	. . . . . . . . . . . 12 ⊢ (⟨p, ⟨z, x⟩⟩ ∈ ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ¬ ((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z))
80	79	exbii 1582	. . . . . . . . . . 11 ⊢ (∃p⟨p, ⟨z, x⟩⟩ ∈ ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ∃p ¬ ((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z))
81		exnal 1574	. . . . . . . . . . 11 ⊢ (∃p ¬ ((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z) ↔ ¬ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z))
82	13, 80, 81	3bitrri 263	. . . . . . . . . 10 ⊢ (¬ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z) ↔ ⟨z, x⟩ ∈ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ))
83	82	con1bii 321	. . . . . . . . 9 ⊢ (¬ ⟨z, x⟩ ∈ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z))
84	12, 83	bitri 240	. . . . . . . 8 ⊢ (⟨z, x⟩ ∈ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ↔ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z))
85	9, 84	anbi12i 678	. . . . . . 7 ⊢ ((⟨z, {y}⟩ ∈ ◡ S ∧ ⟨z, x⟩ ∈ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ↔ (y ∈ z ∧ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)))
86	2, 85	bitri 240	. . . . . 6 ⊢ (⟨z, ⟨{y}, x⟩⟩ ∈ (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ↔ (y ∈ z ∧ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)))
87	86	exbii 1582	. . . . 5 ⊢ (∃z⟨z, ⟨{y}, x⟩⟩ ∈ (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ↔ ∃z(y ∈ z ∧ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)))
88		elrn2 4898	. . . . 5 ⊢ (⟨{y}, x⟩ ∈ ran (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ↔ ∃z⟨z, ⟨{y}, x⟩⟩ ∈ (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )))
89		df-tc 6104	. . . . . . . 8 ⊢ Tc ∪x = (℩p(p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q))
90		dfiota2 4341	. . . . . . . 8 ⊢ (℩p(p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q)) = ∪{z ∣ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)}
91	89, 90	eqtri 2373	. . . . . . 7 ⊢ Tc ∪x = ∪{z ∣ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)}
92	91	eleq2i 2417	. . . . . 6 ⊢ (y ∈ Tc ∪x ↔ y ∈ ∪{z ∣ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)})
93		eluniab 3904	. . . . . 6 ⊢ (y ∈ ∪{z ∣ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)} ↔ ∃z(y ∈ z ∧ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)))
94	92, 93	bitri 240	. . . . 5 ⊢ (y ∈ Tc ∪x ↔ ∃z(y ∈ z ∧ ∀p((p ∈ NC ∧ ∃q ∈ ∪xp = Nc ℘1q) ↔ p = z)))
95	87, 88, 94	3bitr4i 268	. . . 4 ⊢ (⟨{y}, x⟩ ∈ ran (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ↔ y ∈ Tc ∪x)
96	95	releqmpt 5809	. . 3 ⊢ ((1c × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ran (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ))) “ 1c)) = (x ∈ 1c ↦ Tc ∪x)
97	1, 96	eqtr4i 2376	. 2 ⊢ TcFn = ((1c × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ran (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ))) “ 1c))
98		1cex 4143	. . 3 ⊢ 1c ∈ V
99		ssetex 4745	. . . . . 6 ⊢ S ∈ V
100	99	cnvex 5103	. . . . 5 ⊢ ◡ S ∈ V
101		ncsex 6112	. . . . . . . . . . 11 ⊢ NC ∈ V
102		vvex 4110	. . . . . . . . . . 11 ⊢ V ∈ V
103	101, 102	xpex 5116	. . . . . . . . . 10 ⊢ ( NC × V) ∈ V
104		pw1fnex 5853	. . . . . . . . . . . . . 14 ⊢ Pw1Fn ∈ V
105	104	siex 4754	. . . . . . . . . . . . 13 ⊢ SI Pw1Fn ∈ V
106	99, 105	coex 4751	. . . . . . . . . . . 12 ⊢ ( S ∘ SI Pw1Fn ) ∈ V
107	100	siex 4754	. . . . . . . . . . . . . 14 ⊢ SI ◡ S ∈ V
108	107, 99	txpex 5786	. . . . . . . . . . . . 13 ⊢ ( SI ◡ S ⊗ S ) ∈ V
109	108, 98	imaex 4748	. . . . . . . . . . . 12 ⊢ (( SI ◡ S ⊗ S ) “ 1c) ∈ V
110	106, 109	txpex 5786	. . . . . . . . . . 11 ⊢ (( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) ∈ V
111	98	pw1ex 4304	. . . . . . . . . . 11 ⊢ ℘11c ∈ V
112	110, 111	imaex 4748	. . . . . . . . . 10 ⊢ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c) ∈ V
113	103, 112	inex 4106	. . . . . . . . 9 ⊢ (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ∈ V
114	113	ins2ex 5798	. . . . . . . 8 ⊢ Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ∈ V
115		idex 5505	. . . . . . . . 9 ⊢ I ∈ V
116	115	ins3ex 5799	. . . . . . . 8 ⊢ Ins3 I ∈ V
117	114, 116	symdifex 4109	. . . . . . 7 ⊢ ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ∈ V
118	117	rnex 5108	. . . . . 6 ⊢ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ∈ V
119	118	complex 4105	. . . . 5 ⊢ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ) ∈ V
120	100, 119	txpex 5786	. . . 4 ⊢ (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ∈ V
121	120	rnex 5108	. . 3 ⊢ ran (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I )) ∈ V
122	98, 121	mptexlem 5811	. 2 ⊢ ((1c × V) ∩ ◡ ∼ (( Ins3 S ⊕ Ins2 ran (◡ S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S ∘ SI Pw1Fn ) ⊗ (( SI ◡ S ⊗ S ) “ 1c)) “ ℘11c)) ⊕ Ins3 I ))) “ 1c)) ∈ V
123	97, 122	eqeltri 2423	1 ⊢ TcFn ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∩ cin 3209   ⊕ csymdif 3210  {csn 3738  ∪cuni 3892  1_cc1c 4135  ℘₁cpw1 4136  ℩cio 4338  ⟨cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   ∘ ccom 4722   “ cima 4723   I cid 4764   × cxp 4771  ^◡ccnv 4772  ran crn 4774   ↦ cmpt 5652   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   Pw1Fn cpw1fn 5766   NC cncs 6089   Nc cnc 6092   T_c ctc 6094  TcFnctcfn 6098

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-13 1712  ax-14 1714  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-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  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-pss 3262  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-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-mpt 5653  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-tc 6104  df-tcfn 6108

This theorem is referenced by:  nmembers1lem1  6269  nchoicelem11  6300  nchoicelem16  6305