Theorem foundex 5915

Description: The class of all founded relationships is a set. (Contributed by SF, 19-Feb-2015.)

Assertion

Ref	Expression
foundex	|- Fr e. _V

Proof of Theorem foundex

Dummy variables a r t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-found 5906	. . 3 |- Fr = {<.r, a>. | A.x((x C_ a /\ x =/= (/)) -> E.z e. x A.y e. x (yrz -> y = z))}
2		vex 2863	. . . . . . 7 |- r e. _V
3		vex 2863	. . . . . . 7 |- a e. _V
4	2, 3	opex 4589	. . . . . 6 |- <.r, a>. e. _V
5	4	elcompl 3226	. . . . 5 |- (<.r, a>. e. ∼ ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> -. <.r, a>. e. ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))))
6		elrn2 4898	. . . . . . 7 |- (<.r, a>. e. ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> E.x<.x, <.r, a>.>. e. ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))))
7		oteltxp 5783	. . . . . . . . 9 |- (<.x, <.r, a>.>. e. ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> (<.x, r>. e. ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) /\ <.x, a>. e. ( S i^i ( ∼ {(/)} X. _V))))
8		vex 2863	. . . . . . . . . . . . 13 |- x e. _V
9	8, 2	opex 4589	. . . . . . . . . . . 12 |- <.x, r>. e. _V
10	9	elcompl 3226	. . . . . . . . . . 11 |- (<.x, r>. e. ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) <-> -. <.x, r>. e. (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c))
11		elin 3220	. . . . . . . . . . . . . . 15 |- (<.{z}, <.x, r>.>. e. ( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)) <-> (<.{z}, <.x, r>.>. e. Ins3 S /\ <.{z}, <.x, r>.>. e. ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)))
12	2	otelins3 5793	. . . . . . . . . . . . . . . . 17 |- (<.{z}, <.x, r>.>. e. Ins3 S <-> <.{z}, x>. e. S )
13		vex 2863	. . . . . . . . . . . . . . . . . 18 |- z e. _V
14	13, 8	opelssetsn 4761	. . . . . . . . . . . . . . . . 17 |- (<.{z}, x>. e. S <-> z e. x)
15	12, 14	bitri 240	. . . . . . . . . . . . . . . 16 |- (<.{z}, <.x, r>.>. e. Ins3 S <-> z e. x)
16		snex 4112	. . . . . . . . . . . . . . . . . . 19 |- {z} e. _V
17	16, 9	opex 4589	. . . . . . . . . . . . . . . . . 18 |- <.{z}, <.x, r>.>. e. _V
18	17	elcompl 3226	. . . . . . . . . . . . . . . . 17 |- (<.{z}, <.x, r>.>. e. ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) <-> -. <.{z}, <.x, r>.>. e. (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))
19		elin 3220	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.{y}, <.{z}, <.x, r>.>.>. e. ( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )) <-> (<.{y}, <.{z}, <.x, r>.>.>. e. Ins2 Ins3 S /\ <.{y}, <.{z}, <.x, r>.>.>. e. ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )))
20	16	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{y}, <.{z}, <.x, r>.>.>. e. Ins2 Ins3 S <-> <.{y}, <.x, r>.>. e. Ins3 S )
21	2	otelins3 5793	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{y}, <.x, r>.>. e. Ins3 S <-> <.{y}, x>. e. S )
22		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- y e. _V
23	22, 8	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{y}, x>. e. S <-> y e. x)
24	20, 21, 23	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{y}, <.{z}, <.x, r>.>.>. e. Ins2 Ins3 S <-> y e. x)
25		eldif 3222	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (<.{y}, <.{z}, <.x, r>.>.>. e. ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ) <-> (<.{y}, <.{z}, <.x, r>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) /\ -. <.{y}, <.{z}, <.x, r>.>.>. e. Ins3 _I ))
26		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S ) <-> (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins4 SI3 _I /\ <.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins2 Ins2 Ins2 S ))
27	9	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins4 SI3 _I <-> <.{t}, <.{y}, {z}>.>. e. SI3 _I )
28		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- t e. _V
29	28, 22, 13	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{t}, <.{y}, {z}>.>. e. SI3 _I <-> <.t, <.y, z>.>. e. _I )
30		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (t _I <.y, z>. <-> <.t, <.y, z>.>. e. _I )
31	22, 13	opex 4589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- <.y, z>. e. _V
32	31	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (t _I <.y, z>. <-> t = <.y, z>.)
33	30, 32	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.t, <.y, z>.>. e. _I <-> t = <.y, z>.)
34	27, 29, 33	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins4 SI3 _I <-> t = <.y, z>.)
35		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- {y} e. _V
36	35	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins2 Ins2 Ins2 S <-> <.{t}, <.{z}, <.x, r>.>.>. e. Ins2 Ins2 S )
37	16	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{t}, <.{z}, <.x, r>.>.>. e. Ins2 Ins2 S <-> <.{t}, <.x, r>.>. e. Ins2 S )
38	8	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{t}, <.x, r>.>. e. Ins2 S <-> <.{t}, r>. e. S )
39	28, 2	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (<.{t}, r>. e. S <-> t e. r)
40	38, 39	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (<.{t}, <.x, r>.>. e. Ins2 S <-> t e. r)
41	36, 37, 40	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins2 Ins2 Ins2 S <-> t e. r)
42	34, 41	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins4 SI3 _I /\ <.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. Ins2 Ins2 Ins2 S ) <-> (t = <.y, z>. /\ t e. r))
43	26, 42	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S ) <-> (t = <.y, z>. /\ t e. r))
44	43	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (E.t<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S ) <-> E.t(t = <.y, z>. /\ t e. r))
45		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{y}, <.{z}, <.x, r>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) <-> E.t<.{t}, <.{y}, <.{z}, <.x, r>.>.>.>. e. ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S ))
46		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (yrz <-> <.y, z>. e. r)
47		df-clel 2349	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.y, z>. e. r <-> E.t(t = <.y, z>. /\ t e. r))
48	46, 47	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (yrz <-> E.t(t = <.y, z>. /\ t e. r))
49	44, 45, 48	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (<.{y}, <.{z}, <.x, r>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) <-> yrz)
50	9	otelins3 5793	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.{y}, <.{z}, <.x, r>.>.>. e. Ins3 _I <-> <.{y}, {z}>. e. _I )
51		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ({y} _I {z} <-> <.{y}, {z}>. e. _I )
52	16	ideq 4871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ({y} _I {z} <-> {y} = {z})
53	22	sneqb 3877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ({y} = {z} <-> y = z)
54	52, 53	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ({y} _I {z} <-> y = z)
55	51, 54	bitr3i 242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (<.{y}, {z}>. e. _I <-> y = z)
56	50, 55	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (<.{y}, <.{z}, <.x, r>.>.>. e. Ins3 _I <-> y = z)
57	56	notbii 287	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (-. <.{y}, <.{z}, <.x, r>.>.>. e. Ins3 _I <-> -. y = z)
58	49, 57	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((<.{y}, <.{z}, <.x, r>.>.>. e. (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) /\ -. <.{y}, <.{z}, <.x, r>.>.>. e. Ins3 _I ) <-> (yrz /\ -. y = z))
59	25, 58	bitri 240	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.{y}, <.{z}, <.x, r>.>.>. e. ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ) <-> (yrz /\ -. y = z))
60	24, 59	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . 22 |- ((<.{y}, <.{z}, <.x, r>.>.>. e. Ins2 Ins3 S /\ <.{y}, <.{z}, <.x, r>.>.>. e. ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )) <-> (y e. x /\ (yrz /\ -. y = z)))
61	19, 60	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (<.{y}, <.{z}, <.x, r>.>.>. e. ( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )) <-> (y e. x /\ (yrz /\ -. y = z)))
62	61	exbii 1582	. . . . . . . . . . . . . . . . . . . 20 |- (E.y<.{y}, <.{z}, <.x, r>.>.>. e. ( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )) <-> E.y(y e. x /\ (yrz /\ -. y = z)))
63		elima1c 4948	. . . . . . . . . . . . . . . . . . . 20 |- (<.{z}, <.x, r>.>. e. (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) <-> E.y<.{y}, <.{z}, <.x, r>.>.>. e. ( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )))
64		df-rex 2621	. . . . . . . . . . . . . . . . . . . 20 |- (E.y e. x (yrz /\ -. y = z) <-> E.y(y e. x /\ (yrz /\ -. y = z)))
65	62, 63, 64	3bitr4i 268	. . . . . . . . . . . . . . . . . . 19 |- (<.{z}, <.x, r>.>. e. (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) <-> E.y e. x (yrz /\ -. y = z))
66		rexanali 2661	. . . . . . . . . . . . . . . . . . 19 |- (E.y e. x (yrz /\ -. y = z) <-> -. A.y e. x (yrz -> y = z))
67	65, 66	bitri 240	. . . . . . . . . . . . . . . . . 18 |- (<.{z}, <.x, r>.>. e. (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) <-> -. A.y e. x (yrz -> y = z))
68	67	con2bii 322	. . . . . . . . . . . . . . . . 17 |- (A.y e. x (yrz -> y = z) <-> -. <.{z}, <.x, r>.>. e. (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))
69	18, 68	bitr4i 243	. . . . . . . . . . . . . . . 16 |- (<.{z}, <.x, r>.>. e. ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) <-> A.y e. x (yrz -> y = z))
70	15, 69	anbi12i 678	. . . . . . . . . . . . . . 15 |- ((<.{z}, <.x, r>.>. e. Ins3 S /\ <.{z}, <.x, r>.>. e. ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)) <-> (z e. x /\ A.y e. x (yrz -> y = z)))
71	11, 70	bitri 240	. . . . . . . . . . . . . 14 |- (<.{z}, <.x, r>.>. e. ( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)) <-> (z e. x /\ A.y e. x (yrz -> y = z)))
72	71	exbii 1582	. . . . . . . . . . . . 13 |- (E.z<.{z}, <.x, r>.>. e. ( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)) <-> E.z(z e. x /\ A.y e. x (yrz -> y = z)))
73		elima1c 4948	. . . . . . . . . . . . 13 |- (<.x, r>. e. (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) <-> E.z<.{z}, <.x, r>.>. e. ( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)))
74		df-rex 2621	. . . . . . . . . . . . 13 |- (E.z e. x A.y e. x (yrz -> y = z) <-> E.z(z e. x /\ A.y e. x (yrz -> y = z)))
75	72, 73, 74	3bitr4i 268	. . . . . . . . . . . 12 |- (<.x, r>. e. (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) <-> E.z e. x A.y e. x (yrz -> y = z))
76	75	notbii 287	. . . . . . . . . . 11 |- (-. <.x, r>. e. (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) <-> -. E.z e. x A.y e. x (yrz -> y = z))
77	10, 76	bitri 240	. . . . . . . . . 10 |- (<.x, r>. e. ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) <-> -. E.z e. x A.y e. x (yrz -> y = z))
78		elin 3220	. . . . . . . . . . 11 |- (<.x, a>. e. ( S i^i ( ∼ {(/)} X. _V)) <-> (<.x, a>. e. S /\ <.x, a>. e. ( ∼ {(/)} X. _V)))
79		df-br 4641	. . . . . . . . . . . . 13 |- (x S a <-> <.x, a>. e. S )
80	8, 3	brsset 4759	. . . . . . . . . . . . 13 |- (x S a <-> x C_ a)
81	79, 80	bitr3i 242	. . . . . . . . . . . 12 |- (<.x, a>. e. S <-> x C_ a)
82		opelxp 4812	. . . . . . . . . . . . . 14 |- (<.x, a>. e. ( ∼ {(/)} X. _V) <-> (x e. ∼ {(/)} /\ a e. _V))
83	3, 82	mpbiran2 885	. . . . . . . . . . . . 13 |- (<.x, a>. e. ( ∼ {(/)} X. _V) <-> x e. ∼ {(/)})
84	8	elcompl 3226	. . . . . . . . . . . . 13 |- (x e. ∼ {(/)} <-> -. x e. {(/)})
85		elsn 3749	. . . . . . . . . . . . . 14 |- (x e. {(/)} <-> x = (/))
86	85	necon3bbii 2548	. . . . . . . . . . . . 13 |- (-. x e. {(/)} <-> x =/= (/))
87	83, 84, 86	3bitri 262	. . . . . . . . . . . 12 |- (<.x, a>. e. ( ∼ {(/)} X. _V) <-> x =/= (/))
88	81, 87	anbi12i 678	. . . . . . . . . . 11 |- ((<.x, a>. e. S /\ <.x, a>. e. ( ∼ {(/)} X. _V)) <-> (x C_ a /\ x =/= (/)))
89	78, 88	bitri 240	. . . . . . . . . 10 |- (<.x, a>. e. ( S i^i ( ∼ {(/)} X. _V)) <-> (x C_ a /\ x =/= (/)))
90	77, 89	anbi12ci 679	. . . . . . . . 9 |- ((<.x, r>. e. ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) /\ <.x, a>. e. ( S i^i ( ∼ {(/)} X. _V))) <-> ((x C_ a /\ x =/= (/)) /\ -. E.z e. x A.y e. x (yrz -> y = z)))
91	7, 90	bitri 240	. . . . . . . 8 |- (<.x, <.r, a>.>. e. ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> ((x C_ a /\ x =/= (/)) /\ -. E.z e. x A.y e. x (yrz -> y = z)))
92	91	exbii 1582	. . . . . . 7 |- (E.x<.x, <.r, a>.>. e. ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> E.x((x C_ a /\ x =/= (/)) /\ -. E.z e. x A.y e. x (yrz -> y = z)))
93		exanali 1585	. . . . . . 7 |- (E.x((x C_ a /\ x =/= (/)) /\ -. E.z e. x A.y e. x (yrz -> y = z)) <-> -. A.x((x C_ a /\ x =/= (/)) -> E.z e. x A.y e. x (yrz -> y = z)))
94	6, 92, 93	3bitri 262	. . . . . 6 |- (<.r, a>. e. ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> -. A.x((x C_ a /\ x =/= (/)) -> E.z e. x A.y e. x (yrz -> y = z)))
95	94	con2bii 322	. . . . 5 |- (A.x((x C_ a /\ x =/= (/)) -> E.z e. x A.y e. x (yrz -> y = z)) <-> -. <.r, a>. e. ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))))
96	5, 95	bitr4i 243	. . . 4 |- (<.r, a>. e. ∼ ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) <-> A.x((x C_ a /\ x =/= (/)) -> E.z e. x A.y e. x (yrz -> y = z)))
97	96	opabbi2i 4867	. . 3 |- ∼ ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) = {<.r, a>. | A.x((x C_ a /\ x =/= (/)) -> E.z e. x A.y e. x (yrz -> y = z))}
98	1, 97	eqtr4i 2376	. 2 |- Fr = ∼ ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V)))
99		ssetex 4745	. . . . . . . . 9 |- S e. _V
100	99	ins3ex 5799	. . . . . . . 8 |- Ins3 S e. _V
101	100	ins2ex 5798	. . . . . . . . . . 11 |- Ins2 Ins3 S e. _V
102		idex 5505	. . . . . . . . . . . . . . . 16 |- _I e. _V
103	102	si3ex 5807	. . . . . . . . . . . . . . 15 |- SI3 _I e. _V
104	103	ins4ex 5800	. . . . . . . . . . . . . 14 |- Ins4 SI3 _I e. _V
105	99	ins2ex 5798	. . . . . . . . . . . . . . . 16 |- Ins2 S e. _V
106	105	ins2ex 5798	. . . . . . . . . . . . . . 15 |- Ins2 Ins2 S e. _V
107	106	ins2ex 5798	. . . . . . . . . . . . . 14 |- Ins2 Ins2 Ins2 S e. _V
108	104, 107	inex 4106	. . . . . . . . . . . . 13 |- ( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S ) e. _V
109		1cex 4143	. . . . . . . . . . . . 13 |- 1c e. _V
110	108, 109	imaex 4748	. . . . . . . . . . . 12 |- (( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) e. _V
111	102	ins3ex 5799	. . . . . . . . . . . 12 |- Ins3 _I e. _V
112	110, 111	difex 4108	. . . . . . . . . . 11 |- ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ) e. _V
113	101, 112	inex 4106	. . . . . . . . . 10 |- ( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I )) e. _V
114	113, 109	imaex 4748	. . . . . . . . 9 |- (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) e. _V
115	114	complex 4105	. . . . . . . 8 |- ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c) e. _V
116	100, 115	inex 4106	. . . . . . 7 |- ( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c)) e. _V
117	116, 109	imaex 4748	. . . . . 6 |- (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) e. _V
118	117	complex 4105	. . . . 5 |- ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) e. _V
119		snex 4112	. . . . . . . 8 |- {(/)} e. _V
120	119	complex 4105	. . . . . . 7 |- ∼ {(/)} e. _V
121		vvex 4110	. . . . . . 7 |- _V e. _V
122	120, 121	xpex 5116	. . . . . 6 |- ( ∼ {(/)} X. _V) e. _V
123	99, 122	inex 4106	. . . . 5 |- ( S i^i ( ∼ {(/)} X. _V)) e. _V
124	118, 123	txpex 5786	. . . 4 |- ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) e. _V
125	124	rnex 5108	. . 3 |- ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) e. _V
126	125	complex 4105	. 2 |- ∼ ran ( ∼ (( Ins3 S i^i ∼ (( Ins2 Ins3 S i^i ((( Ins4 SI3 _I i^i Ins2 Ins2 Ins2 S )"1c) \ Ins3 _I ))"1c))"1c) (x) ( S i^i ( ∼ {(/)} X. _V))) e. _V
127	98, 126	eqeltri 2423	1 |- Fr e. _V

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  A.wral 2615  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   \ cdif 3207   i^i cin 3209   C_ wss 3258  (/)c0 3551  {csn 3738  1_cc1c 4135  <.cop 4562  {copab 4623   class class class wbr 4640   S csset 4720  "cima 4723   _I cid 4764   X. cxp 4771  ran crn 4774   (x) ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Fr cfound 5895

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-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-found 5906

This theorem is referenced by:  weex  5920