Theorem tcfnex 6245

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

Assertion

Ref	Expression
tcfnex	|- TcFn e. _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 e. 1c |-> Tc U.x)
2		oteltxp 5783	. . . . . . 7 |- (<.z, <.{y}, x>.>. e. (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) <-> (<.z, {y}>. e. `' S /\ <.z, x>. e. ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )))
3		df-br 4641	. . . . . . . . 9 |- (z`' S {y} <-> <.z, {y}>. e. `' S )
4		brcnv 4893	. . . . . . . . . 10 |- (z`' S {y} <-> {y} S z)
5		vex 2863	. . . . . . . . . . 11 |- y e. _V
6		vex 2863	. . . . . . . . . . 11 |- z e. _V
7	5, 6	brssetsn 4760	. . . . . . . . . 10 |- ({y} S z <-> y e. z)
8	4, 7	bitri 240	. . . . . . . . 9 |- (z`' S {y} <-> y e. z)
9	3, 8	bitr3i 242	. . . . . . . 8 |- (<.z, {y}>. e. `' S <-> y e. z)
10		vex 2863	. . . . . . . . . . 11 |- x e. _V
11	6, 10	opex 4589	. . . . . . . . . 10 |- <.z, x>. e. _V
12	11	elcompl 3226	. . . . . . . . 9 |- (<.z, x>. e. ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> -. <.z, x>. e. ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ))
13		elrn2 4898	. . . . . . . . . . 11 |- (<.z, x>. e. ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> E.p<.p, <.z, x>.>. e. ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ))
14		elsymdif 3224	. . . . . . . . . . . . 13 |- (<.p, <.z, x>.>. e. ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> -. (<.p, <.z, x>.>. e. Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> <.p, <.z, x>.>. e. Ins3 _I ))
15	6	otelins2 5792	. . . . . . . . . . . . . . 15 |- (<.p, <.z, x>.>. e. Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> <.p, x>. e. (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)))
16		elin 3220	. . . . . . . . . . . . . . 15 |- (<.p, x>. e. (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> (<.p, x>. e. ( NC X. _V) /\ <.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)))
17		opelxp 4812	. . . . . . . . . . . . . . . . . 18 |- (<.p, x>. e. ( NC X. _V) <-> (p e. NC /\ x e. _V))
18	10, 17	mpbiran2 885	. . . . . . . . . . . . . . . . 17 |- (<.p, x>. e. ( NC X. _V) <-> p e. NC )
19	18	anbi1i 676	. . . . . . . . . . . . . . . 16 |- ((<.p, x>. e. ( NC X. _V) /\ <.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> (p e. NC /\ <.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)))
20		ncseqnc 6129	. . . . . . . . . . . . . . . . . . 19 |- (p e. NC -> (p = Nc ~P1 q <-> ~P1 q e. p))
21	20	rexbidv 2636	. . . . . . . . . . . . . . . . . 18 |- (p e. NC -> (E.q e. U.xp = Nc ~P1 q <-> E.q e. U.x~P1 q e. p))
22		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{{q}}, <.p, x>.>. e. (( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c)) <-> (<.{{q}}, p>. e. ( S o. SI Pw1Fn ) /\ <.{{q}}, x>. e. (( SI `' S (x) S )"1c)))
23		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- {q} e. _V
24	23	brsnsi1 5776	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ({{q}} SI Pw1Fn u <-> E.t(u = {t} /\ {q} Pw1Fn t))
25	24	anbi1i 676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (({{q}} SI Pw1Fn u /\ u S p) <-> (E.t(u = {t} /\ {q} Pw1Fn t) /\ u S p))
26		19.41v 1901	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (E.t((u = {t} /\ {q} Pw1Fn t) /\ u S p) <-> (E.t(u = {t} /\ {q} Pw1Fn t) /\ u S p))
27	25, 26	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (({{q}} SI Pw1Fn u /\ u S p) <-> E.t((u = {t} /\ {q} Pw1Fn t) /\ u S p))
28	27	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (E.u({{q}} SI Pw1Fn u /\ u S p) <-> E.uE.t((u = {t} /\ {q} Pw1Fn t) /\ u S p))
29		excom 1741	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (E.uE.t((u = {t} /\ {q} Pw1Fn t) /\ u S p) <-> E.tE.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 |- (E.u((u = {t} /\ {q} Pw1Fn t) /\ u S p) <-> E.u(u = {t} /\ ({q} Pw1Fn t /\ u S p)))
32		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- {t} e. _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 |- (E.u(u = {t} /\ ({q} Pw1Fn t /\ u S p)) <-> ({q} Pw1Fn t /\ {t} S p))
36		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- q e. _V
37	36	brpw1fn 5855	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ({q} Pw1Fn t <-> t = ~P1 q)
38		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- t e. _V
39		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- p e. _V
40	38, 39	brssetsn 4760	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ({t} S p <-> t e. p)
41	37, 40	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (({q} Pw1Fn t /\ {t} S p) <-> (t = ~P1 q /\ t e. p))
42	31, 35, 41	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (E.u((u = {t} /\ {q} Pw1Fn t) /\ u S p) <-> (t = ~P1 q /\ t e. p))
43	42	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (E.tE.u((u = {t} /\ {q} Pw1Fn t) /\ u S p) <-> E.t(t = ~P1 q /\ t e. p))
44	28, 29, 43	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 |- (E.u({{q}} SI Pw1Fn u /\ u S p) <-> E.t(t = ~P1 q /\ t e. p))
45		opelco 4885	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{{q}}, p>. e. ( S o. SI Pw1Fn ) <-> E.u({{q}} SI Pw1Fn u /\ u S p))
46		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . . 23 |- (~P1 q e. p <-> E.t(t = ~P1 q /\ t e. p))
47	44, 45, 46	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{{q}}, p>. e. ( S o. SI Pw1Fn ) <-> ~P1 q e. p)
48		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{t}, <.{{q}}, x>.>. e. ( SI `' S (x) S ) <-> (<.{t}, {{q}}>. e. SI `' S /\ <.{t}, x>. e. S ))
49		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ({t} SI `' S {{q}} <-> <.{t}, {{q}}>. e. 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 e. t)
53	50, 51, 52	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ({t} SI `' S {{q}} <-> q e. t)
54	49, 53	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{t}, {{q}}>. e. SI `' S <-> q e. t)
55	38, 10	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{t}, x>. e. S <-> t e. x)
56	54, 55	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((<.{t}, {{q}}>. e. SI `' S /\ <.{t}, x>. e. S ) <-> (q e. t /\ t e. x))
57	48, 56	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{t}, <.{{q}}, x>.>. e. ( SI `' S (x) S ) <-> (q e. t /\ t e. x))
58	57	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . 23 |- (E.t<.{t}, <.{{q}}, x>.>. e. ( SI `' S (x) S ) <-> E.t(q e. t /\ t e. x))
59		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{{q}}, x>. e. (( SI `' S (x) S )"1c) <-> E.t<.{t}, <.{{q}}, x>.>. e. ( SI `' S (x) S ))
60		eluni 3895	. . . . . . . . . . . . . . . . . . . . . . 23 |- (q e. U.x <-> E.t(q e. t /\ t e. x))
61	58, 59, 60	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{{q}}, x>. e. (( SI `' S (x) S )"1c) <-> q e. U.x)
62	47, 61	anbi12i 678	. . . . . . . . . . . . . . . . . . . . 21 |- ((<.{{q}}, p>. e. ( S o. SI Pw1Fn ) /\ <.{{q}}, x>. e. (( SI `' S (x) S )"1c)) <-> (~P1 q e. p /\ q e. U.x))
63		ancom 437	. . . . . . . . . . . . . . . . . . . . 21 |- ((~P1 q e. p /\ q e. U.x) <-> (q e. U.x /\ ~P1 q e. p))
64	22, 62, 63	3bitri 262	. . . . . . . . . . . . . . . . . . . 20 |- (<.{{q}}, <.p, x>.>. e. (( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c)) <-> (q e. U.x /\ ~P1 q e. p))
65	64	exbii 1582	. . . . . . . . . . . . . . . . . . 19 |- (E.q<.{{q}}, <.p, x>.>. e. (( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c)) <-> E.q(q e. U.x /\ ~P1 q e. p))
66		elimapw11c 4949	. . . . . . . . . . . . . . . . . . 19 |- (<.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c) <-> E.q<.{{q}}, <.p, x>.>. e. (( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c)))
67		df-rex 2621	. . . . . . . . . . . . . . . . . . 19 |- (E.q e. U.x~P1 q e. p <-> E.q(q e. U.x /\ ~P1 q e. p))
68	65, 66, 67	3bitr4i 268	. . . . . . . . . . . . . . . . . 18 |- (<.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c) <-> E.q e. U.x~P1 q e. p)
69	21, 68	syl6rbbr 255	. . . . . . . . . . . . . . . . 17 |- (p e. NC -> (<.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c) <-> E.q e. U.xp = Nc ~P1 q))
70	69	pm5.32i 618	. . . . . . . . . . . . . . . 16 |- ((p e. NC /\ <.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> (p e. NC /\ E.q e. U.xp = Nc ~P1 q))
71	19, 70	bitri 240	. . . . . . . . . . . . . . 15 |- ((<.p, x>. e. ( NC X. _V) /\ <.p, x>. e. ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> (p e. NC /\ E.q e. U.xp = Nc ~P1 q))
72	15, 16, 71	3bitri 262	. . . . . . . . . . . . . 14 |- (<.p, <.z, x>.>. e. Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> (p e. NC /\ E.q e. U.xp = Nc ~P1 q))
73	10	otelins3 5793	. . . . . . . . . . . . . . 15 |- (<.p, <.z, x>.>. e. Ins3 _I <-> <.p, z>. e. _I )
74		df-br 4641	. . . . . . . . . . . . . . . 16 |- (p _I z <-> <.p, z>. e. _I )
75	6	ideq 4871	. . . . . . . . . . . . . . . 16 |- (p _I z <-> p = z)
76	74, 75	bitr3i 242	. . . . . . . . . . . . . . 15 |- (<.p, z>. e. _I <-> p = z)
77	73, 76	bitri 240	. . . . . . . . . . . . . 14 |- (<.p, <.z, x>.>. e. Ins3 _I <-> p = z)
78	72, 77	bibi12i 306	. . . . . . . . . . . . 13 |- ((<.p, <.z, x>.>. e. Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) <-> <.p, <.z, x>.>. e. Ins3 _I ) <-> ((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z))
79	14, 78	xchbinx 301	. . . . . . . . . . . 12 |- (<.p, <.z, x>.>. e. ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> -. ((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z))
80	79	exbii 1582	. . . . . . . . . . 11 |- (E.p<.p, <.z, x>.>. e. ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> E.p -. ((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z))
81		exnal 1574	. . . . . . . . . . 11 |- (E.p -. ((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z) <-> -. A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z))
82	13, 80, 81	3bitrri 263	. . . . . . . . . 10 |- (-. A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z) <-> <.z, x>. e. ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ))
83	82	con1bii 321	. . . . . . . . 9 |- (-. <.z, x>. e. ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z))
84	12, 83	bitri 240	. . . . . . . 8 |- (<.z, x>. e. ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) <-> A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z))
85	9, 84	anbi12i 678	. . . . . . 7 |- ((<.z, {y}>. e. `' S /\ <.z, x>. e. ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) <-> (y e. z /\ A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)))
86	2, 85	bitri 240	. . . . . 6 |- (<.z, <.{y}, x>.>. e. (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) <-> (y e. z /\ A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)))
87	86	exbii 1582	. . . . 5 |- (E.z<.z, <.{y}, x>.>. e. (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) <-> E.z(y e. z /\ A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)))
88		elrn2 4898	. . . . 5 |- (<.{y}, x>. e. ran (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) <-> E.z<.z, <.{y}, x>.>. e. (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )))
89		df-tc 6104	. . . . . . . 8 |- Tc U.x = (iotap(p e. NC /\ E.q e. U.xp = Nc ~P1 q))
90		dfiota2 4341	. . . . . . . 8 |- (iotap(p e. NC /\ E.q e. U.xp = Nc ~P1 q)) = U.{z | A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)}
91	89, 90	eqtri 2373	. . . . . . 7 |- Tc U.x = U.{z | A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)}
92	91	eleq2i 2417	. . . . . 6 |- (y e. Tc U.x <-> y e. U.{z | A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)})
93		eluniab 3904	. . . . . 6 |- (y e. U.{z | A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)} <-> E.z(y e. z /\ A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)))
94	92, 93	bitri 240	. . . . 5 |- (y e. Tc U.x <-> E.z(y e. z /\ A.p((p e. NC /\ E.q e. U.xp = Nc ~P1 q) <-> p = z)))
95	87, 88, 94	3bitr4i 268	. . . 4 |- (<.{y}, x>. e. ran (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) <-> y e. Tc U.x)
96	95	releqmpt 5809	. . 3 |- ((1c X. _V) i^i `' ∼ (( Ins3 S (+) Ins2 ran (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )))"1c)) = (x e. 1c |-> Tc U.x)
97	1, 96	eqtr4i 2376	. 2 |- TcFn = ((1c X. _V) i^i `' ∼ (( Ins3 S (+) Ins2 ran (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )))"1c))
98		1cex 4143	. . 3 |- 1c e. _V
99		ssetex 4745	. . . . . 6 |- S e. _V
100	99	cnvex 5103	. . . . 5 |- `' S e. _V
101		ncsex 6112	. . . . . . . . . . 11 |- NC e. _V
102		vvex 4110	. . . . . . . . . . 11 |- _V e. _V
103	101, 102	xpex 5116	. . . . . . . . . 10 |- ( NC X. _V) e. _V
104		pw1fnex 5853	. . . . . . . . . . . . . 14 |- Pw1Fn e. _V
105	104	siex 4754	. . . . . . . . . . . . 13 |- SI Pw1Fn e. _V
106	99, 105	coex 4751	. . . . . . . . . . . 12 |- ( S o. SI Pw1Fn ) e. _V
107	100	siex 4754	. . . . . . . . . . . . . 14 |- SI `' S e. _V
108	107, 99	txpex 5786	. . . . . . . . . . . . 13 |- ( SI `' S (x) S ) e. _V
109	108, 98	imaex 4748	. . . . . . . . . . . 12 |- (( SI `' S (x) S )"1c) e. _V
110	106, 109	txpex 5786	. . . . . . . . . . 11 |- (( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c)) e. _V
111	98	pw1ex 4304	. . . . . . . . . . 11 |- ~P1 1c e. _V
112	110, 111	imaex 4748	. . . . . . . . . 10 |- ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c) e. _V
113	103, 112	inex 4106	. . . . . . . . 9 |- (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) e. _V
114	113	ins2ex 5798	. . . . . . . 8 |- Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) e. _V
115		idex 5505	. . . . . . . . 9 |- _I e. _V
116	115	ins3ex 5799	. . . . . . . 8 |- Ins3 _I e. _V
117	114, 116	symdifex 4109	. . . . . . 7 |- ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) e. _V
118	117	rnex 5108	. . . . . 6 |- ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) e. _V
119	118	complex 4105	. . . . 5 |- ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I ) e. _V
120	100, 119	txpex 5786	. . . 4 |- (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) e. _V
121	120	rnex 5108	. . 3 |- ran (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )) e. _V
122	98, 121	mptexlem 5811	. 2 |- ((1c X. _V) i^i `' ∼ (( Ins3 S (+) Ins2 ran (`' S (x) ∼ ran ( Ins2 (( NC X. _V) i^i ((( S o. SI Pw1Fn ) (x) (( SI `' S (x) S )"1c))"~P1 1c)) (+) Ins3 _I )))"1c)) e. _V
123	97, 122	eqeltri 2423	1 |- TcFn e. _V

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   i^i cin 3209   (+) csymdif 3210  {csn 3738  U.cuni 3892  1_cc1c 4135  ~P₁ cpw1 4136  iotacio 4338  <.cop 4562   class class class wbr 4640   S csset 4720   SI csi 4721   o. ccom 4722  "cima 4723   _I cid 4764   X. cxp 4771  `'ccnv 4772  ran crn 4774   |-> cmpt 5652   (x) 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