Theorem funsex 5829

Description: The class of all functions forms a set. (Contributed by SF, 18-Feb-2015.)

Assertion

Ref	Expression
funsex	⊢ Funs ∈ V

Proof of Theorem funsex

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

Step	Hyp	Ref	Expression
1		df-funs 5761	. . 3 ⊢ Funs = {f ∣ Fun f}
2		elima1c 4948	. . . . . . . 8 ⊢ (f ∈ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ∃x⟨{x}, f⟩ ∈ ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c))
3		snex 4112	. . . . . . . . . . . 12 ⊢ {x} ∈ V
4		vex 2863	. . . . . . . . . . . 12 ⊢ f ∈ V
5	3, 4	opex 4589	. . . . . . . . . . 11 ⊢ ⟨{x}, f⟩ ∈ V
6	5	elcompl 3226	. . . . . . . . . 10 ⊢ (⟨{x}, f⟩ ∈ ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ↔ ¬ ⟨{x}, f⟩ ∈ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c))
7		elima1c 4948	. . . . . . . . . . . 12 ⊢ (⟨{x}, f⟩ ∈ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ↔ ∃z⟨{z}, ⟨{x}, f⟩⟩ ∈ ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c))
8		elima1c 4948	. . . . . . . . . . . . . . . 16 ⊢ (⟨{z}, ⟨{x}, f⟩⟩ ∈ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ↔ ∃y⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ))
9		eldif 3222	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) ↔ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∧ ¬ ⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins3 I ))
10		snex 4112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {z} ∈ V
11	10	otelins2 5792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ ⟨{y}, ⟨{x}, f⟩⟩ ∈ (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ))
12		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ x ∈ V
13		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ y ∈ V
14	12, 13	opex 4589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨x, y⟩ ∈ V
15	14, 4	opelssetsn 4761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{⟨x, y⟩}, f⟩ ∈ S ↔ ⟨x, y⟩ ∈ f)
16		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (p( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd )⟨{y}, ⟨{x}, f⟩⟩ ↔ ⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ))
17		elin 3220	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) ↔ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∧ ⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 Ins2 2nd ))
18	4	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ↔ ⟨p, ⟨{y}, {x}⟩⟩ ∈ ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c))
19		elima1c 4948	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨p, ⟨{y}, {x}⟩⟩ ∈ ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ↔ ∃q⟨{q}, ⟨p, ⟨{y}, {x}⟩⟩⟩ ∈ (◡1st ⊗ SI3 (2nd ⊗ 1st )))
20		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨{q}, ⟨p, ⟨{y}, {x}⟩⟩⟩ ∈ (◡1st ⊗ SI3 (2nd ⊗ 1st )) ↔ (⟨{q}, p⟩ ∈ ◡1st ∧ ⟨{q}, ⟨{y}, {x}⟩⟩ ∈ SI3 (2nd ⊗ 1st )))
21		opelcnv 4894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨{q}, p⟩ ∈ ◡1st ↔ ⟨p, {q}⟩ ∈ 1st )
22		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (p1st {q} ↔ ⟨p, {q}⟩ ∈ 1st )
23	21, 22	bitr4i 243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨{q}, p⟩ ∈ ◡1st ↔ p1st {q})
24		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ q ∈ V
25	24, 13, 12	otsnelsi3 5806	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨{q}, ⟨{y}, {x}⟩⟩ ∈ SI3 (2nd ⊗ 1st ) ↔ ⟨q, ⟨y, x⟩⟩ ∈ (2nd ⊗ 1st ))
26		oteltxp 5783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟨q, ⟨y, x⟩⟩ ∈ (2nd ⊗ 1st ) ↔ (⟨q, y⟩ ∈ 2nd ∧ ⟨q, x⟩ ∈ 1st ))
27		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (q1st x ↔ ⟨q, x⟩ ∈ 1st )
28		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (q2nd y ↔ ⟨q, y⟩ ∈ 2nd )
29	27, 28	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((q1st x ∧ q2nd y) ↔ (⟨q, x⟩ ∈ 1st ∧ ⟨q, y⟩ ∈ 2nd ))
30		ancom 437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((⟨q, x⟩ ∈ 1st ∧ ⟨q, y⟩ ∈ 2nd ) ↔ (⟨q, y⟩ ∈ 2nd ∧ ⟨q, x⟩ ∈ 1st ))
31	29, 30	bitr2i 241	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((⟨q, y⟩ ∈ 2nd ∧ ⟨q, x⟩ ∈ 1st ) ↔ (q1st x ∧ q2nd y))
32	12, 13	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((q1st x ∧ q2nd y) ↔ q = ⟨x, y⟩)
33	26, 31, 32	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨q, ⟨y, x⟩⟩ ∈ (2nd ⊗ 1st ) ↔ q = ⟨x, y⟩)
34	25, 33	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨{q}, ⟨{y}, {x}⟩⟩ ∈ SI3 (2nd ⊗ 1st ) ↔ q = ⟨x, y⟩)
35	23, 34	anbi12ci 679	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((⟨{q}, p⟩ ∈ ◡1st ∧ ⟨{q}, ⟨{y}, {x}⟩⟩ ∈ SI3 (2nd ⊗ 1st )) ↔ (q = ⟨x, y⟩ ∧ p1st {q}))
36	20, 35	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨{q}, ⟨p, ⟨{y}, {x}⟩⟩⟩ ∈ (◡1st ⊗ SI3 (2nd ⊗ 1st )) ↔ (q = ⟨x, y⟩ ∧ p1st {q}))
37	36	exbii 1582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃q⟨{q}, ⟨p, ⟨{y}, {x}⟩⟩⟩ ∈ (◡1st ⊗ SI3 (2nd ⊗ 1st )) ↔ ∃q(q = ⟨x, y⟩ ∧ p1st {q}))
38		sneq 3745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (q = ⟨x, y⟩ → {q} = {⟨x, y⟩})
39	38	breq2d 4652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (q = ⟨x, y⟩ → (p1st {q} ↔ p1st {⟨x, y⟩}))
40	14, 39	ceqsexv 2895	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃q(q = ⟨x, y⟩ ∧ p1st {q}) ↔ p1st {⟨x, y⟩})
41	19, 37, 40	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨p, ⟨{y}, {x}⟩⟩ ∈ ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ↔ p1st {⟨x, y⟩})
42	18, 41	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ↔ p1st {⟨x, y⟩})
43	3	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨p, ⟨{x}, f⟩⟩ ∈ Ins2 2nd ↔ ⟨p, f⟩ ∈ 2nd )
44		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ {y} ∈ V
45	44	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 Ins2 2nd ↔ ⟨p, ⟨{x}, f⟩⟩ ∈ Ins2 2nd )
46		df-br 4641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (p2nd f ↔ ⟨p, f⟩ ∈ 2nd )
47	43, 45, 46	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 Ins2 2nd ↔ p2nd f)
48	42, 47	anbi12i 678	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∧ ⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 Ins2 2nd ) ↔ (p1st {⟨x, y⟩} ∧ p2nd f))
49	17, 48	bitri 240	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨p, ⟨{y}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) ↔ (p1st {⟨x, y⟩} ∧ p2nd f))
50		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {⟨x, y⟩} ∈ V
51	50, 4	op1st2nd 5791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((p1st {⟨x, y⟩} ∧ p2nd f) ↔ p = ⟨{⟨x, y⟩}, f⟩)
52	16, 49, 51	3bitri 262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (p( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd )⟨{y}, ⟨{x}, f⟩⟩ ↔ p = ⟨{⟨x, y⟩}, f⟩)
53	52	rexbii 2640	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃p ∈ S p( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd )⟨{y}, ⟨{x}, f⟩⟩ ↔ ∃p ∈ S p = ⟨{⟨x, y⟩}, f⟩)
54		elima 4755	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{y}, ⟨{x}, f⟩⟩ ∈ (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ ∃p ∈ S p( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd )⟨{y}, ⟨{x}, f⟩⟩)
55		risset 2662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨{⟨x, y⟩}, f⟩ ∈ S ↔ ∃p ∈ S p = ⟨{⟨x, y⟩}, f⟩)
56	53, 54, 55	3bitr4i 268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{y}, ⟨{x}, f⟩⟩ ∈ (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ ⟨{⟨x, y⟩}, f⟩ ∈ S )
57		df-br 4641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (xfy ↔ ⟨x, y⟩ ∈ f)
58	15, 56, 57	3bitr4i 268	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{x}, f⟩⟩ ∈ (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ xfy)
59	11, 58	bitri 240	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ xfy)
60	5	otelins3 5793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins3 I ↔ ⟨{y}, {z}⟩ ∈ I )
61	10	ideq 4871	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({y} I {z} ↔ {y} = {z})
62		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({y} I {z} ↔ ⟨{y}, {z}⟩ ∈ I )
63	13	sneqb 3877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({y} = {z} ↔ y = z)
64	61, 62, 63	3bitr3i 266	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨{y}, {z}⟩ ∈ I ↔ y = z)
65	60, 64	bitri 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins3 I ↔ y = z)
66	65	notbii 287	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ ⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins3 I ↔ ¬ y = z)
67	59, 66	anbi12i 678	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∧ ¬ ⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ Ins3 I ) ↔ (xfy ∧ ¬ y = z))
68	9, 67	bitri 240	. . . . . . . . . . . . . . . . 17 ⊢ (⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) ↔ (xfy ∧ ¬ y = z))
69	68	exbii 1582	. . . . . . . . . . . . . . . 16 ⊢ (∃y⟨{y}, ⟨{z}, ⟨{x}, f⟩⟩⟩ ∈ ( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) ↔ ∃y(xfy ∧ ¬ y = z))
70	8, 69	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨{z}, ⟨{x}, f⟩⟩ ∈ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ↔ ∃y(xfy ∧ ¬ y = z))
71	70	notbii 287	. . . . . . . . . . . . . 14 ⊢ (¬ ⟨{z}, ⟨{x}, f⟩⟩ ∈ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ↔ ¬ ∃y(xfy ∧ ¬ y = z))
72	10, 5	opex 4589	. . . . . . . . . . . . . . 15 ⊢ ⟨{z}, ⟨{x}, f⟩⟩ ∈ V
73	72	elcompl 3226	. . . . . . . . . . . . . 14 ⊢ (⟨{z}, ⟨{x}, f⟩⟩ ∈ ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ↔ ¬ ⟨{z}, ⟨{x}, f⟩⟩ ∈ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c))
74		exanali 1585	. . . . . . . . . . . . . . 15 ⊢ (∃y(xfy ∧ ¬ y = z) ↔ ¬ ∀y(xfy → y = z))
75	74	con2bii 322	. . . . . . . . . . . . . 14 ⊢ (∀y(xfy → y = z) ↔ ¬ ∃y(xfy ∧ ¬ y = z))
76	71, 73, 75	3bitr4i 268	. . . . . . . . . . . . 13 ⊢ (⟨{z}, ⟨{x}, f⟩⟩ ∈ ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ↔ ∀y(xfy → y = z))
77	76	exbii 1582	. . . . . . . . . . . 12 ⊢ (∃z⟨{z}, ⟨{x}, f⟩⟩ ∈ ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ↔ ∃z∀y(xfy → y = z))
78	7, 77	bitri 240	. . . . . . . . . . 11 ⊢ (⟨{x}, f⟩ ∈ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ↔ ∃z∀y(xfy → y = z))
79	78	notbii 287	. . . . . . . . . 10 ⊢ (¬ ⟨{x}, f⟩ ∈ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ↔ ¬ ∃z∀y(xfy → y = z))
80	6, 79	bitri 240	. . . . . . . . 9 ⊢ (⟨{x}, f⟩ ∈ ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ↔ ¬ ∃z∀y(xfy → y = z))
81	80	exbii 1582	. . . . . . . 8 ⊢ (∃x⟨{x}, f⟩ ∈ ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ↔ ∃x ¬ ∃z∀y(xfy → y = z))
82	2, 81	bitri 240	. . . . . . 7 ⊢ (f ∈ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ∃x ¬ ∃z∀y(xfy → y = z))
83	82	notbii 287	. . . . . 6 ⊢ (¬ f ∈ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ¬ ∃x ¬ ∃z∀y(xfy → y = z))
84	4	elcompl 3226	. . . . . 6 ⊢ (f ∈ ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ¬ f ∈ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c))
85		alex 1572	. . . . . 6 ⊢ (∀x∃z∀y(xfy → y = z) ↔ ¬ ∃x ¬ ∃z∀y(xfy → y = z))
86	83, 84, 85	3bitr4i 268	. . . . 5 ⊢ (f ∈ ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ∀x∃z∀y(xfy → y = z))
87		dffun3 5121	. . . . 5 ⊢ (Fun f ↔ ∀x∃z∀y(xfy → y = z))
88	86, 87	bitr4i 243	. . . 4 ⊢ (f ∈ ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ Fun f)
89	88	eqabi 2465	. . 3 ⊢ ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) = {f ∣ Fun f}
90	1, 89	eqtr4i 2376	. 2 ⊢ Funs = ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c)
91		1stex 4740	. . . . . . . . . . . . . . . 16 ⊢ 1st ∈ V
92	91	cnvex 5103	. . . . . . . . . . . . . . 15 ⊢ ◡1st ∈ V
93		2ndex 5113	. . . . . . . . . . . . . . . . 17 ⊢ 2nd ∈ V
94	93, 91	txpex 5786	. . . . . . . . . . . . . . . 16 ⊢ (2nd ⊗ 1st ) ∈ V
95	94	si3ex 5807	. . . . . . . . . . . . . . 15 ⊢ SI3 (2nd ⊗ 1st ) ∈ V
96	92, 95	txpex 5786	. . . . . . . . . . . . . 14 ⊢ (◡1st ⊗ SI3 (2nd ⊗ 1st )) ∈ V
97		1cex 4143	. . . . . . . . . . . . . 14 ⊢ 1c ∈ V
98	96, 97	imaex 4748	. . . . . . . . . . . . 13 ⊢ ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∈ V
99	98	ins4ex 5800	. . . . . . . . . . . 12 ⊢ Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∈ V
100	93	ins2ex 5798	. . . . . . . . . . . . 13 ⊢ Ins2 2nd ∈ V
101	100	ins2ex 5798	. . . . . . . . . . . 12 ⊢ Ins2 Ins2 2nd ∈ V
102	99, 101	inex 4106	. . . . . . . . . . 11 ⊢ ( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) ∈ V
103		ssetex 4745	. . . . . . . . . . 11 ⊢ S ∈ V
104	102, 103	imaex 4748	. . . . . . . . . 10 ⊢ (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∈ V
105	104	ins2ex 5798	. . . . . . . . 9 ⊢ Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∈ V
106		idex 5505	. . . . . . . . . 10 ⊢ I ∈ V
107	106	ins3ex 5799	. . . . . . . . 9 ⊢ Ins3 I ∈ V
108	105, 107	difex 4108	. . . . . . . 8 ⊢ ( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) ∈ V
109	108, 97	imaex 4748	. . . . . . 7 ⊢ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ∈ V
110	109	complex 4105	. . . . . 6 ⊢ ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) ∈ V
111	110, 97	imaex 4748	. . . . 5 ⊢ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ∈ V
112	111	complex 4105	. . . 4 ⊢ ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) ∈ V
113	112, 97	imaex 4748	. . 3 ⊢ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ∈ V
114	113	complex 4105	. 2 ⊢ ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((◡1st ⊗ SI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ∖ Ins3 I ) “ 1c) “ 1c) “ 1c) ∈ V
115	90, 114	eqeltri 2423	1 ⊢ Funs ∈ V

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209  {csn 3738  1_cc1c 4135  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   S csset 4720   “ cima 4723   I cid 4764  ^◡ccnv 4772  Fun wfun 4776  2^nd c2nd 4784   ⊗ ctxp 5736   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Funs cfuns 5760

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-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 proof 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-cnv 4786  df-fun 4790  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-ins4 5757  df-si3 5759  df-funs 5761

This theorem is used by:  fnsex  5833  mapexi  6004  pmex  6006  fnpm  6009