NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  funsex GIF version

Theorem funsex 5828
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.
StepHypRef Expression
1 df-funs 5760 . . 3 Funs = {f Fun f}
2 elima1c 4947 . . . . . . . 8 (f ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ x{x}, f ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c))
3 snex 4111 . . . . . . . . . . . 12 {x} V
4 vex 2862 . . . . . . . . . . . 12 f V
53, 4opex 4588 . . . . . . . . . . 11 {x}, f V
65elcompl 3225 . . . . . . . . . 10 ({x}, f ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) ↔ ¬ {x}, f ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c))
7 elima1c 4947 . . . . . . . . . . . 12 ({x}, f ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) ↔ z{z}, {x}, f ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c))
8 elima1c 4947 . . . . . . . . . . . . . . . 16 ({z}, {x}, f (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) ↔ y{y}, {z}, {x}, f ( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ))
9 eldif 3221 . . . . . . . . . . . . . . . . . 18 ({y}, {z}, {x}, f ( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) ↔ ({y}, {z}, {x}, f Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ¬ {y}, {z}, {x}, f Ins3 I ))
10 snex 4111 . . . . . . . . . . . . . . . . . . . . 21 {z} V
1110otelins2 5791 . . . . . . . . . . . . . . . . . . . 20 ({y}, {z}, {x}, f Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ {y}, {x}, f (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ))
12 vex 2862 . . . . . . . . . . . . . . . . . . . . . . 23 x V
13 vex 2862 . . . . . . . . . . . . . . . . . . . . . . 23 y V
1412, 13opex 4588 . . . . . . . . . . . . . . . . . . . . . 22 x, y V
1514, 4opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . 21 ({x, y}, f S x, y f)
16 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . 24 (p( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ){y}, {x}, fp, {y}, {x}, f ( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ))
17 elin 3219 . . . . . . . . . . . . . . . . . . . . . . . . 25 (p, {y}, {x}, f ( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) ↔ (p, {y}, {x}, f Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) p, {y}, {x}, f Ins2 Ins2 2nd ))
184oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p, {y}, {x}, f Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ↔ p, {y}, {x} ((1stSI3 (2nd ⊗ 1st )) “ 1c))
19 elima1c 4947 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (p, {y}, {x} ((1stSI3 (2nd ⊗ 1st )) “ 1c) ↔ q{q}, p, {y}, {x} (1stSI3 (2nd ⊗ 1st )))
20 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ({q}, p, {y}, {x} (1stSI3 (2nd ⊗ 1st )) ↔ ({q}, p 1st {q}, {y}, {x} SI3 (2nd ⊗ 1st )))
21 opelcnv 4893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({q}, p 1stp, {q} 1st )
22 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (p1st {q} ↔ p, {q} 1st )
2321, 22bitr4i 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({q}, p 1stp1st {q})
24 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 q V
2524, 13, 12otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ({q}, {y}, {x} SI3 (2nd ⊗ 1st ) ↔ q, y, x (2nd ⊗ 1st ))
26 oteltxp 5782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (q, y, x (2nd ⊗ 1st ) ↔ (q, y 2nd q, x 1st ))
27 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (q1st xq, x 1st )
28 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (q2nd yq, y 2nd )
2927, 28anbi12i 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 ))
3129, 30bitr2i 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((q, y 2nd q, x 1st ) ↔ (q1st x q2nd y))
3212, 13op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((q1st x q2nd y) ↔ q = x, y)
3326, 31, 323bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (q, y, x (2nd ⊗ 1st ) ↔ q = x, y)
3425, 33bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ({q}, {y}, {x} SI3 (2nd ⊗ 1st ) ↔ q = x, y)
3523, 34anbi12ci 679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (({q}, p 1st {q}, {y}, {x} SI3 (2nd ⊗ 1st )) ↔ (q = x, y p1st {q}))
3620, 35bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ({q}, p, {y}, {x} (1stSI3 (2nd ⊗ 1st )) ↔ (q = x, y p1st {q}))
3736exbii 1582 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (q{q}, p, {y}, {x} (1stSI3 (2nd ⊗ 1st )) ↔ q(q = x, y p1st {q}))
38 sneq 3744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (q = x, y → {q} = {x, y})
3938breq2d 4651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (q = x, y → (p1st {q} ↔ p1st {x, y}))
4014, 39ceqsexv 2894 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (q(q = x, y p1st {q}) ↔ p1st {x, y})
4119, 37, 403bitri 262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p, {y}, {x} ((1stSI3 (2nd ⊗ 1st )) “ 1c) ↔ p1st {x, y})
4218, 41bitri 240 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p, {y}, {x}, f Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ↔ p1st {x, y})
433otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p, {x}, f Ins2 2ndp, f 2nd )
44 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 {y} V
4544otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p, {y}, {x}, f Ins2 Ins2 2ndp, {x}, f Ins2 2nd )
46 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (p2nd fp, f 2nd )
4743, 45, 463bitr4i 268 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (p, {y}, {x}, f Ins2 Ins2 2ndp2nd f)
4842, 47anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((p, {y}, {x}, f Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) p, {y}, {x}, f Ins2 Ins2 2nd ) ↔ (p1st {x, y} p2nd f))
4917, 48bitri 240 . . . . . . . . . . . . . . . . . . . . . . . 24 (p, {y}, {x}, f ( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) ↔ (p1st {x, y} p2nd f))
50 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . . 25 {x, y} V
5150, 4op1st2nd 5790 . . . . . . . . . . . . . . . . . . . . . . . 24 ((p1st {x, y} p2nd f) ↔ p = {x, y}, f)
5216, 49, 513bitri 262 . . . . . . . . . . . . . . . . . . . . . . 23 (p( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ){y}, {x}, fp = {x, y}, f)
5352rexbii 2639 . . . . . . . . . . . . . . . . . . . . . 22 (p S p( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ){y}, {x}, fp S p = {x, y}, f)
54 elima 4754 . . . . . . . . . . . . . . . . . . . . . 22 ({y}, {x}, f (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ p S p( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ){y}, {x}, f)
55 risset 2661 . . . . . . . . . . . . . . . . . . . . . 22 ({x, y}, f S p S p = {x, y}, f)
5653, 54, 553bitr4i 268 . . . . . . . . . . . . . . . . . . . . 21 ({y}, {x}, f (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ {x, y}, f S )
57 df-br 4640 . . . . . . . . . . . . . . . . . . . . 21 (xfyx, y f)
5815, 56, 573bitr4i 268 . . . . . . . . . . . . . . . . . . . 20 ({y}, {x}, f (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ xfy)
5911, 58bitri 240 . . . . . . . . . . . . . . . . . . 19 ({y}, {z}, {x}, f Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ↔ xfy)
605otelins3 5792 . . . . . . . . . . . . . . . . . . . . 21 ({y}, {z}, {x}, f Ins3 I ↔ {y}, {z} I )
6110ideq 4870 . . . . . . . . . . . . . . . . . . . . . 22 ({y} I {z} ↔ {y} = {z})
62 df-br 4640 . . . . . . . . . . . . . . . . . . . . . 22 ({y} I {z} ↔ {y}, {z} I )
6313sneqb 3876 . . . . . . . . . . . . . . . . . . . . . 22 ({y} = {z} ↔ y = z)
6461, 62, 633bitr3i 266 . . . . . . . . . . . . . . . . . . . . 21 ({y}, {z} I ↔ y = z)
6560, 64bitri 240 . . . . . . . . . . . . . . . . . . . 20 ({y}, {z}, {x}, f Ins3 I ↔ y = z)
6665notbii 287 . . . . . . . . . . . . . . . . . . 19 {y}, {z}, {x}, f Ins3 I ↔ ¬ y = z)
6759, 66anbi12i 678 . . . . . . . . . . . . . . . . . 18 (({y}, {z}, {x}, f Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) ¬ {y}, {z}, {x}, f Ins3 I ) ↔ (xfy ¬ y = z))
689, 67bitri 240 . . . . . . . . . . . . . . . . 17 ({y}, {z}, {x}, f ( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) ↔ (xfy ¬ y = z))
6968exbii 1582 . . . . . . . . . . . . . . . 16 (y{y}, {z}, {x}, f ( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) ↔ y(xfy ¬ y = z))
708, 69bitri 240 . . . . . . . . . . . . . . 15 ({z}, {x}, f (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) ↔ y(xfy ¬ y = z))
7170notbii 287 . . . . . . . . . . . . . 14 {z}, {x}, f (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) ↔ ¬ y(xfy ¬ y = z))
7210, 5opex 4588 . . . . . . . . . . . . . . 15 {z}, {x}, f V
7372elcompl 3225 . . . . . . . . . . . . . 14 ({z}, {x}, f ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) ↔ ¬ {z}, {x}, f (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c))
74 exanali 1585 . . . . . . . . . . . . . . 15 (y(xfy ¬ y = z) ↔ ¬ y(xfyy = z))
7574con2bii 322 . . . . . . . . . . . . . 14 (y(xfyy = z) ↔ ¬ y(xfy ¬ y = z))
7671, 73, 753bitr4i 268 . . . . . . . . . . . . 13 ({z}, {x}, f ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) ↔ y(xfyy = z))
7776exbii 1582 . . . . . . . . . . . 12 (z{z}, {x}, f ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) ↔ zy(xfyy = z))
787, 77bitri 240 . . . . . . . . . . 11 ({x}, f ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) ↔ zy(xfyy = z))
7978notbii 287 . . . . . . . . . 10 {x}, f ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) ↔ ¬ zy(xfyy = z))
806, 79bitri 240 . . . . . . . . 9 ({x}, f ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) ↔ ¬ zy(xfyy = z))
8180exbii 1582 . . . . . . . 8 (x{x}, f ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) ↔ x ¬ zy(xfyy = z))
822, 81bitri 240 . . . . . . 7 (f ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ x ¬ zy(xfyy = z))
8382notbii 287 . . . . . 6 f ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ¬ x ¬ zy(xfyy = z))
844elcompl 3225 . . . . . 6 (f ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ ¬ f ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c))
85 alex 1572 . . . . . 6 (xzy(xfyy = z) ↔ ¬ x ¬ zy(xfyy = z))
8683, 84, 853bitr4i 268 . . . . 5 (f ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ xzy(xfyy = z))
87 dffun3 5120 . . . . 5 (Fun fxzy(xfyy = z))
8886, 87bitr4i 243 . . . 4 (f ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) ↔ Fun f)
8988abbi2i 2464 . . 3 ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) = {f Fun f}
901, 89eqtr4i 2376 . 2 Funs = ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c)
91 1stex 4739 . . . . . . . . . . . . . . . 16 1st V
9291cnvex 5102 . . . . . . . . . . . . . . 15 1st V
93 2ndex 5112 . . . . . . . . . . . . . . . . 17 2nd V
9493, 91txpex 5785 . . . . . . . . . . . . . . . 16 (2nd ⊗ 1st ) V
9594si3ex 5806 . . . . . . . . . . . . . . 15 SI3 (2nd ⊗ 1st ) V
9692, 95txpex 5785 . . . . . . . . . . . . . 14 (1stSI3 (2nd ⊗ 1st )) V
97 1cex 4142 . . . . . . . . . . . . . 14 1c V
9896, 97imaex 4747 . . . . . . . . . . . . 13 ((1stSI3 (2nd ⊗ 1st )) “ 1c) V
9998ins4ex 5799 . . . . . . . . . . . 12 Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) V
10093ins2ex 5797 . . . . . . . . . . . . 13 Ins2 2nd V
101100ins2ex 5797 . . . . . . . . . . . 12 Ins2 Ins2 2nd V
10299, 101inex 4105 . . . . . . . . . . 11 ( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) V
103 ssetex 4744 . . . . . . . . . . 11 S V
104102, 103imaex 4747 . . . . . . . . . 10 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) V
105104ins2ex 5797 . . . . . . . . 9 Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) V
106 idex 5504 . . . . . . . . . 10 I V
107106ins3ex 5798 . . . . . . . . 9 Ins3 I V
108105, 107difex 4107 . . . . . . . 8 ( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) V
109108, 97imaex 4747 . . . . . . 7 (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) V
110109complex 4104 . . . . . 6 ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) V
111110, 97imaex 4747 . . . . 5 ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) V
112111complex 4104 . . . 4 ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) V
113112, 97imaex 4747 . . 3 ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) V
114113complex 4104 . 2 ∼ ( ∼ ( ∼ (( Ins2 (( Ins4 ((1stSI3 (2nd ⊗ 1st )) “ 1c) ∩ Ins2 Ins2 2nd ) “ S ) Ins3 I ) “ 1c) “ 1c) “ 1c) V
11590, 114eqeltri 2423 1 Funs V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   wa 358  wal 1540  wex 1541   = wceq 1642   wcel 1710  {cab 2339  wrex 2615  Vcvv 2859  ccompl 3205   cdif 3206  cin 3208  {csn 3737  1cc1c 4134  cop 4561   class class class wbr 4639  1st c1st 4717   S csset 4719  cima 4722   I cid 4763  ccnv 4771  Fun wfun 4775  2nd c2nd 4783  ctxp 5735   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757   Funs cfuns 5759
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 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-cnv 4785  df-fun 4789  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758  df-funs 5760
This theorem is referenced by:  fnsex  5832  mapexi  6003  pmex  6005  fnpm  6008
  Copyright terms: Public domain W3C validator