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Theorem tcfnex 6244
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.
StepHypRef Expression
1 df-tcfn 6107 . . 3 TcFn = (x 1c Tc x)
2 oteltxp 5782 . . . . . . 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 4640 . . . . . . . . 9 (z S {y} ↔ z, {y} S )
4 brcnv 4892 . . . . . . . . . 10 (z S {y} ↔ {y} S z)
5 vex 2862 . . . . . . . . . . 11 y V
6 vex 2862 . . . . . . . . . . 11 z V
75, 6brssetsn 4759 . . . . . . . . . 10 ({y} S zy z)
84, 7bitri 240 . . . . . . . . 9 (z S {y} ↔ y z)
93, 8bitr3i 242 . . . . . . . 8 (z, {y} S y z)
10 vex 2862 . . . . . . . . . . 11 x V
116, 10opex 4588 . . . . . . . . . 10 z, x V
1211elcompl 3225 . . . . . . . . 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 4897 . . . . . . . . . . 11 (z, x ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I ) ↔ pp, z, x ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I ))
14 elsymdif 3223 . . . . . . . . . . . . 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 ))
156otelins2 5791 . . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . . . . 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 4811 . . . . . . . . . . . . . . . . . 18 (p, x ( NC × V) ↔ (p NC x V))
1810, 17mpbiran2 885 . . . . . . . . . . . . . . . . 17 (p, x ( NC × V) ↔ p NC )
1918anbi1i 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 6128 . . . . . . . . . . . . . . . . . . 19 (p NC → (p = Nc 1q1q p))
2120rexbidv 2635 . . . . . . . . . . . . . . . . . 18 (p NC → (q xp = Nc 1qq x1q p))
22 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . 21 ({{q}}, p, x (( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) ↔ ({{q}}, p ( S SI Pw1Fn ) {{q}}, x (( SI S S ) “ 1c)))
23 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {q} V
2423brsnsi1 5775 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({{q}} SI Pw1Fn ut(u = {t} {q} Pw1Fn t))
2524anbi1i 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))
2725, 26bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({{q}} SI Pw1Fn u u S p) ↔ t((u = {t} {q} Pw1Fn t) u S p))
2827exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . 24 (u({{q}} SI Pw1Fn u u S p) ↔ ut((u = {t} {q} Pw1Fn t) u S p))
29 excom 1741 . . . . . . . . . . . . . . . . . . . . . . . 24 (ut((u = {t} {q} Pw1Fn t) u S p) ↔ tu((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)))
3130exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (u((u = {t} {q} Pw1Fn t) u S p) ↔ u(u = {t} ({q} Pw1Fn t u S p)))
32 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 {t} V
33 breq1 4642 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (u = {t} → (u S p ↔ {t} S p))
3433anbi2d 684 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (u = {t} → (({q} Pw1Fn t u S p) ↔ ({q} Pw1Fn t {t} S p)))
3532, 34ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (u(u = {t} ({q} Pw1Fn t u S p)) ↔ ({q} Pw1Fn t {t} S p))
36 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 q V
3736brpw1fn 5854 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({q} Pw1Fn tt = 1q)
38 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 t V
39 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 p V
4038, 39brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({t} S pt p)
4137, 40anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (({q} Pw1Fn t {t} S p) ↔ (t = 1q t p))
4231, 35, 413bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . 25 (u((u = {t} {q} Pw1Fn t) u S p) ↔ (t = 1q t p))
4342exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . 24 (tu((u = {t} {q} Pw1Fn t) u S p) ↔ t(t = 1q t p))
4428, 29, 433bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 (u({{q}} SI Pw1Fn u u S p) ↔ t(t = 1q t p))
45 opelco 4884 . . . . . . . . . . . . . . . . . . . . . . 23 ({{q}}, p ( S SI Pw1Fn ) ↔ u({{q}} SI Pw1Fn u u S p))
46 df-clel 2349 . . . . . . . . . . . . . . . . . . . . . . 23 (1q pt(t = 1q t p))
4744, 45, 463bitr4i 268 . . . . . . . . . . . . . . . . . . . . . 22 ({{q}}, p ( S SI Pw1Fn ) ↔ 1q p)
48 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({t}, {{q}}, x ( SI S S ) ↔ ({t}, {{q}} SI S {t}, x S ))
49 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({t} SI S {{q}} ↔ {t}, {{q}} SI S )
5038, 23brsnsi 5773 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ({t} SI S {{q}} ↔ t S {q})
51 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (t S {q} ↔ {q} S t)
5236, 38brssetsn 4759 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ({q} S tq t)
5350, 51, 523bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({t} SI S {{q}} ↔ q t)
5449, 53bitr3i 242 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({t}, {{q}} SI S q t)
5538, 10opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({t}, x S t x)
5654, 55anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({t}, {{q}} SI S {t}, x S ) ↔ (q t t x))
5748, 56bitri 240 . . . . . . . . . . . . . . . . . . . . . . . 24 ({t}, {{q}}, x ( SI S S ) ↔ (q t t x))
5857exbii 1582 . . . . . . . . . . . . . . . . . . . . . . 23 (t{t}, {{q}}, x ( SI S S ) ↔ t(q t t x))
59 elima1c 4947 . . . . . . . . . . . . . . . . . . . . . . 23 ({{q}}, x (( SI S S ) “ 1c) ↔ t{t}, {{q}}, x ( SI S S ))
60 eluni 3894 . . . . . . . . . . . . . . . . . . . . . . 23 (q xt(q t t x))
6158, 59, 603bitr4i 268 . . . . . . . . . . . . . . . . . . . . . 22 ({{q}}, x (( SI S S ) “ 1c) ↔ q x)
6247, 61anbi12i 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))
6422, 62, 633bitri 262 . . . . . . . . . . . . . . . . . . . 20 ({{q}}, p, x (( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) ↔ (q x 1q p))
6564exbii 1582 . . . . . . . . . . . . . . . . . . 19 (q{{q}}, p, x (( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) ↔ q(q x 1q p))
66 elimapw11c 4948 . . . . . . . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . . . . . . . 19 (q x1q pq(q x 1q p))
6865, 66, 673bitr4i 268 . . . . . . . . . . . . . . . . . 18 (p, x ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c) ↔ q x1q p)
6921, 68syl6rbbr 255 . . . . . . . . . . . . . . . . 17 (p NC → (p, x ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c) ↔ q xp = Nc 1q))
7069pm5.32i 618 . . . . . . . . . . . . . . . 16 ((p NC p, x ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ↔ (p NC q xp = Nc 1q))
7119, 70bitri 240 . . . . . . . . . . . . . . 15 ((p, x ( NC × V) p, x ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ↔ (p NC q xp = Nc 1q))
7215, 16, 713bitri 262 . . . . . . . . . . . . . 14 (p, z, x Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ↔ (p NC q xp = Nc 1q))
7310otelins3 5792 . . . . . . . . . . . . . . 15 (p, z, x Ins3 I ↔ p, z I )
74 df-br 4640 . . . . . . . . . . . . . . . 16 (p I zp, z I )
756ideq 4870 . . . . . . . . . . . . . . . 16 (p I zp = z)
7674, 75bitr3i 242 . . . . . . . . . . . . . . 15 (p, z I ↔ p = z)
7773, 76bitri 240 . . . . . . . . . . . . . 14 (p, z, x Ins3 I ↔ p = z)
7872, 77bibi12i 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))
7914, 78xchbinx 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))
8079exbii 1582 . . . . . . . . . . 11 (pp, 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))
8213, 80, 813bitrri 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 ))
8382con1bii 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))
8412, 83bitri 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))
859, 84anbi12i 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)))
862, 85bitri 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)))
8786exbii 1582 . . . . 5 (zz, {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 4897 . . . . 5 ({y}, x ran ( S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I )) ↔ zz, {y}, x ( S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I )))
89 df-tc 6103 . . . . . . . 8 Tc x = (℩p(p NC q xp = Nc 1q))
90 dfiota2 4340 . . . . . . . 8 (℩p(p NC q xp = Nc 1q)) = {z p((p NC q xp = Nc 1q) ↔ p = z)}
9189, 90eqtri 2373 . . . . . . 7 Tc x = {z p((p NC q xp = Nc 1q) ↔ p = z)}
9291eleq2i 2417 . . . . . 6 (y Tc xy {z p((p NC q xp = Nc 1q) ↔ p = z)})
93 eluniab 3903 . . . . . 6 (y {z p((p NC q xp = Nc 1q) ↔ p = z)} ↔ z(y z p((p NC q xp = Nc 1q) ↔ p = z)))
9492, 93bitri 240 . . . . 5 (y Tc xz(y z p((p NC q xp = Nc 1q) ↔ p = z)))
9587, 88, 943bitr4i 268 . . . 4 ({y}, x ran ( S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I )) ↔ y Tc x)
9695releqmpt 5808 . . 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)
971, 96eqtr4i 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 4142 . . 3 1c V
99 ssetex 4744 . . . . . 6 S V
10099cnvex 5102 . . . . 5 S V
101 ncsex 6111 . . . . . . . . . . 11 NC V
102 vvex 4109 . . . . . . . . . . 11 V V
103101, 102xpex 5115 . . . . . . . . . 10 ( NC × V) V
104 pw1fnex 5852 . . . . . . . . . . . . . 14 Pw1Fn V
105104siex 4753 . . . . . . . . . . . . 13 SI Pw1Fn V
10699, 105coex 4750 . . . . . . . . . . . 12 ( S SI Pw1Fn ) V
107100siex 4753 . . . . . . . . . . . . . 14 SI S V
108107, 99txpex 5785 . . . . . . . . . . . . 13 ( SI S S ) V
109108, 98imaex 4747 . . . . . . . . . . . 12 (( SI S S ) “ 1c) V
110106, 109txpex 5785 . . . . . . . . . . 11 (( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) V
11198pw1ex 4303 . . . . . . . . . . 11 11c V
112110, 111imaex 4747 . . . . . . . . . 10 ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c) V
113103, 112inex 4105 . . . . . . . . 9 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) V
114113ins2ex 5797 . . . . . . . 8 Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) V
115 idex 5504 . . . . . . . . 9 I V
116115ins3ex 5798 . . . . . . . 8 Ins3 I V
117114, 116symdifex 4108 . . . . . . 7 ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I ) V
118117rnex 5107 . . . . . 6 ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I ) V
119118complex 4104 . . . . 5 ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I ) V
120100, 119txpex 5785 . . . 4 ( S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I )) V
121120rnex 5107 . . 3 ran ( S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I )) V
12298, 121mptexlem 5810 . 2 ((1c × V) ∩ ∼ (( Ins3 S Ins2 ran ( S ⊗ ∼ ran ( Ins2 (( NC × V) ∩ ((( S SI Pw1Fn ) ⊗ (( SI S S ) “ 1c)) “ 11c)) ⊕ Ins3 I ))) “ 1c)) V
12397, 122eqeltri 2423 1 TcFn V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 176   wa 358  wal 1540  wex 1541   = wceq 1642   wcel 1710  {cab 2339  wrex 2615  Vcvv 2859  ccompl 3205  cin 3208  csymdif 3209  {csn 3737  cuni 3891  1cc1c 4134  1cpw1 4135  cio 4337  cop 4561   class class class wbr 4639   S csset 4719   SI csi 4720   ccom 4721  cima 4722   I cid 4763   × cxp 4770  ccnv 4771  ran crn 4773   cmpt 5651  ctxp 5735   Ins2 cins2 5749   Ins3 cins3 5751   Pw1Fn cpw1fn 5765   NC cncs 6088   Nc cnc 6091   Tc ctc 6093  TcFnctcfn 6097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 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-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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  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-pss 3261  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-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-mpt 5652  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103  df-tcfn 6107
This theorem is referenced by:  nmembers1lem1  6268  nchoicelem11  6299  nchoicelem16  6304
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