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Theorem composeex 5820
 Description: The compose function is a set. (Contributed by Scott Fenton, 19-Apr-2021.)
Assertion
Ref Expression
composeex Compose V

Proof of Theorem composeex
Dummy variables x y z w t u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-compose 5748 . . 3 Compose = (x V, y V (x y))
2 elopab 4696 . . . . 5 (z {w, t u(wyu uxt)} ↔ wt(z = w, t u(wyu uxt)))
3 df-co 4726 . . . . . 6 (x y) = {w, t u(wyu uxt)}
43eleq2i 2417 . . . . 5 (z (x y) ↔ z {w, t u(wyu uxt)})
5 elima1c 4947 . . . . . 6 ({z}, x, y ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c) ↔ w{w}, {z}, x, y (( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c))
6 elima1c 4947 . . . . . . . 8 ({w}, {z}, x, y (( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) ↔ t{t}, {w}, {z}, x, y ( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)))
7 elin 3219 . . . . . . . . . 10 ({t}, {w}, {z}, x, y ( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) ↔ ({t}, {w}, {z}, x, y Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) {t}, {w}, {z}, x, y (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)))
8 vex 2862 . . . . . . . . . . . . . 14 x V
9 vex 2862 . . . . . . . . . . . . . 14 y V
108, 9opex 4588 . . . . . . . . . . . . 13 x, y V
1110oqelins4 5794 . . . . . . . . . . . 12 ({t}, {w}, {z}, x, y Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ↔ {t}, {w}, {z} SI3 ((V × 1st ) ∩ Ins2 2nd ))
12 vex 2862 . . . . . . . . . . . . 13 t V
13 vex 2862 . . . . . . . . . . . . 13 w V
14 vex 2862 . . . . . . . . . . . . 13 z V
1512, 13, 14otsnelsi3 5805 . . . . . . . . . . . 12 ({t}, {w}, {z} SI3 ((V × 1st ) ∩ Ins2 2nd ) ↔ t, w, z ((V × 1st ) ∩ Ins2 2nd ))
16 elin 3219 . . . . . . . . . . . . 13 (t, w, z ((V × 1st ) ∩ Ins2 2nd ) ↔ (t, w, z (V × 1st ) t, w, z Ins2 2nd ))
17 opelxp 4811 . . . . . . . . . . . . . . . 16 (t, w, z (V × 1st ) ↔ (t V w, z 1st ))
1812, 17mpbiran 884 . . . . . . . . . . . . . . 15 (t, w, z (V × 1st ) ↔ w, z 1st )
19 df-br 4640 . . . . . . . . . . . . . . 15 (w1st zw, z 1st )
20 brcnv 4892 . . . . . . . . . . . . . . 15 (w1st zz1st w)
2118, 19, 203bitr2i 264 . . . . . . . . . . . . . 14 (t, w, z (V × 1st ) ↔ z1st w)
2213otelins2 5791 . . . . . . . . . . . . . . 15 (t, w, z Ins2 2ndt, z 2nd )
23 df-br 4640 . . . . . . . . . . . . . . 15 (t2nd zt, z 2nd )
24 brcnv 4892 . . . . . . . . . . . . . . 15 (t2nd zz2nd t)
2522, 23, 243bitr2i 264 . . . . . . . . . . . . . 14 (t, w, z Ins2 2ndz2nd t)
2621, 25anbi12i 678 . . . . . . . . . . . . 13 ((t, w, z (V × 1st ) t, w, z Ins2 2nd ) ↔ (z1st w z2nd t))
2713, 12op1st2nd 5790 . . . . . . . . . . . . 13 ((z1st w z2nd t) ↔ z = w, t)
2816, 26, 273bitri 262 . . . . . . . . . . . 12 (t, w, z ((V × 1st ) ∩ Ins2 2nd ) ↔ z = w, t)
2911, 15, 283bitri 262 . . . . . . . . . . 11 ({t}, {w}, {z}, x, y Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ↔ z = w, t)
30 elima1c 4947 . . . . . . . . . . . 12 ({t}, {w}, {z}, x, y (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c) ↔ u{u}, {t}, {w}, {z}, x, y ( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)))
31 elin 3219 . . . . . . . . . . . . . 14 ({u}, {t}, {w}, {z}, x, y ( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ ({u}, {t}, {w}, {z}, x, y Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) {u}, {t}, {w}, {z}, x, y (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)))
32 snex 4111 . . . . . . . . . . . . . . . . 17 {t} V
3332otelins2 5791 . . . . . . . . . . . . . . . 16 ({u}, {t}, {w}, {z}, x, y Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ↔ {u}, {w}, {z}, x, y (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c))
34 df-clel 2349 . . . . . . . . . . . . . . . . 17 (w, u yt(t = w, u t y))
35 df-br 4640 . . . . . . . . . . . . . . . . 17 (wyuw, u y)
36 elima1c 4947 . . . . . . . . . . . . . . . . . 18 ({u}, {w}, {z}, x, y (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ↔ t{t}, {u}, {w}, {z}, x, y ( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ))
37 elin 3219 . . . . . . . . . . . . . . . . . . . 20 ({t}, {u}, {w}, {z}, x, y ( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) ↔ ({t}, {u}, {w}, {z}, x, y Ins4 SI3 Swap {t}, {u}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 S ))
38 snex 4111 . . . . . . . . . . . . . . . . . . . . . . . 24 {z} V
3938, 10opex 4588 . . . . . . . . . . . . . . . . . . . . . . 23 {z}, x, y V
4039oqelins4 5794 . . . . . . . . . . . . . . . . . . . . . 22 ({t}, {u}, {w}, {z}, x, y Ins4 SI3 Swap {t}, {u}, {w} SI3 Swap )
41 vex 2862 . . . . . . . . . . . . . . . . . . . . . . . 24 u V
4212, 41, 13otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . . 23 ({t}, {u}, {w} SI3 Swap t, u, w Swap )
43 df-br 4640 . . . . . . . . . . . . . . . . . . . . . . 23 (t Swap u, wt, u, w Swap )
4441, 13brswap2 4860 . . . . . . . . . . . . . . . . . . . . . . 23 (t Swap u, wt = w, u)
4542, 43, 443bitr2i 264 . . . . . . . . . . . . . . . . . . . . . 22 ({t}, {u}, {w} SI3 Swap t = w, u)
4640, 45bitri 240 . . . . . . . . . . . . . . . . . . . . 21 ({t}, {u}, {w}, {z}, x, y Ins4 SI3 Swap t = w, u)
47 snex 4111 . . . . . . . . . . . . . . . . . . . . . . 23 {u} V
4847otelins2 5791 . . . . . . . . . . . . . . . . . . . . . 22 ({t}, {u}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 S {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 S )
49 snex 4111 . . . . . . . . . . . . . . . . . . . . . . 23 {w} V
5049otelins2 5791 . . . . . . . . . . . . . . . . . . . . . 22 ({t}, {w}, {z}, x, y Ins2 Ins2 Ins2 S {t}, {z}, x, y Ins2 Ins2 S )
5138otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . 23 ({t}, {z}, x, y Ins2 Ins2 S {t}, x, y Ins2 S )
528otelins2 5791 . . . . . . . . . . . . . . . . . . . . . . 23 ({t}, x, y Ins2 S {t}, y S )
5312, 9opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . 23 ({t}, y S t y)
5451, 52, 533bitri 262 . . . . . . . . . . . . . . . . . . . . . 22 ({t}, {z}, x, y Ins2 Ins2 S t y)
5548, 50, 543bitri 262 . . . . . . . . . . . . . . . . . . . . 21 ({t}, {u}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 S t y)
5646, 55anbi12i 678 . . . . . . . . . . . . . . . . . . . 20 (({t}, {u}, {w}, {z}, x, y Ins4 SI3 Swap {t}, {u}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 S ) ↔ (t = w, u t y))
5737, 56bitri 240 . . . . . . . . . . . . . . . . . . 19 ({t}, {u}, {w}, {z}, x, y ( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) ↔ (t = w, u t y))
5857exbii 1582 . . . . . . . . . . . . . . . . . 18 (t{t}, {u}, {w}, {z}, x, y ( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) ↔ t(t = w, u t y))
5936, 58bitri 240 . . . . . . . . . . . . . . . . 17 ({u}, {w}, {z}, x, y (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ↔ t(t = w, u t y))
6034, 35, 593bitr4ri 269 . . . . . . . . . . . . . . . 16 ({u}, {w}, {z}, x, y (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ↔ wyu)
6133, 60bitri 240 . . . . . . . . . . . . . . 15 ({u}, {t}, {w}, {z}, x, y Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ↔ wyu)
62 df-clel 2349 . . . . . . . . . . . . . . . 16 (u, t xv(v = u, t v x))
63 df-br 4640 . . . . . . . . . . . . . . . 16 (uxtu, t x)
64 elima1c 4947 . . . . . . . . . . . . . . . . 17 ({u}, {t}, {w}, {z}, x, y (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c) ↔ v{v}, {u}, {t}, {w}, {z}, x, y ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ))
65 elin 3219 . . . . . . . . . . . . . . . . . . 19 ({v}, {u}, {t}, {w}, {z}, x, y ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) ↔ ({v}, {u}, {t}, {w}, {z}, x, y Ins4 SI3 I {v}, {u}, {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 Ins3 S ))
6649, 39opex 4588 . . . . . . . . . . . . . . . . . . . . . 22 {w}, {z}, x, y V
6766oqelins4 5794 . . . . . . . . . . . . . . . . . . . . 21 ({v}, {u}, {t}, {w}, {z}, x, y Ins4 SI3 I ↔ {v}, {u}, {t} SI3 I )
68 vex 2862 . . . . . . . . . . . . . . . . . . . . . . 23 v V
6968, 41, 12otsnelsi3 5805 . . . . . . . . . . . . . . . . . . . . . 22 ({v}, {u}, {t} SI3 I ↔ v, u, t I )
70 df-br 4640 . . . . . . . . . . . . . . . . . . . . . 22 (v I u, tv, u, t I )
7141, 12opex 4588 . . . . . . . . . . . . . . . . . . . . . . 23 u, t V
7271ideq 4870 . . . . . . . . . . . . . . . . . . . . . 22 (v I u, tv = u, t)
7369, 70, 723bitr2i 264 . . . . . . . . . . . . . . . . . . . . 21 ({v}, {u}, {t} SI3 I ↔ v = u, t)
7467, 73bitri 240 . . . . . . . . . . . . . . . . . . . 20 ({v}, {u}, {t}, {w}, {z}, x, y Ins4 SI3 I ↔ v = u, t)
7547otelins2 5791 . . . . . . . . . . . . . . . . . . . . 21 ({v}, {u}, {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 Ins3 S {v}, {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins3 S )
7632otelins2 5791 . . . . . . . . . . . . . . . . . . . . 21 ({v}, {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins3 S {v}, {w}, {z}, x, y Ins2 Ins2 Ins3 S )
7749otelins2 5791 . . . . . . . . . . . . . . . . . . . . . 22 ({v}, {w}, {z}, x, y Ins2 Ins2 Ins3 S {v}, {z}, x, y Ins2 Ins3 S )
7838otelins2 5791 . . . . . . . . . . . . . . . . . . . . . 22 ({v}, {z}, x, y Ins2 Ins3 S {v}, x, y Ins3 S )
799otelins3 5792 . . . . . . . . . . . . . . . . . . . . . . 23 ({v}, x, y Ins3 S {v}, x S )
8068, 8opelssetsn 4760 . . . . . . . . . . . . . . . . . . . . . . 23 ({v}, x S v x)
8179, 80bitri 240 . . . . . . . . . . . . . . . . . . . . . 22 ({v}, x, y Ins3 S v x)
8277, 78, 813bitri 262 . . . . . . . . . . . . . . . . . . . . 21 ({v}, {w}, {z}, x, y Ins2 Ins2 Ins3 S v x)
8375, 76, 823bitri 262 . . . . . . . . . . . . . . . . . . . 20 ({v}, {u}, {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 Ins3 S v x)
8474, 83anbi12i 678 . . . . . . . . . . . . . . . . . . 19 (({v}, {u}, {t}, {w}, {z}, x, y Ins4 SI3 I {v}, {u}, {t}, {w}, {z}, x, y Ins2 Ins2 Ins2 Ins2 Ins3 S ) ↔ (v = u, t v x))
8565, 84bitri 240 . . . . . . . . . . . . . . . . . 18 ({v}, {u}, {t}, {w}, {z}, x, y ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) ↔ (v = u, t v x))
8685exbii 1582 . . . . . . . . . . . . . . . . 17 (v{v}, {u}, {t}, {w}, {z}, x, y ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) ↔ v(v = u, t v x))
8764, 86bitri 240 . . . . . . . . . . . . . . . 16 ({u}, {t}, {w}, {z}, x, y (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c) ↔ v(v = u, t v x))
8862, 63, 873bitr4ri 269 . . . . . . . . . . . . . . 15 ({u}, {t}, {w}, {z}, x, y (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c) ↔ uxt)
8961, 88anbi12i 678 . . . . . . . . . . . . . 14 (({u}, {t}, {w}, {z}, x, y Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) {u}, {t}, {w}, {z}, x, y (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ (wyu uxt))
9031, 89bitri 240 . . . . . . . . . . . . 13 ({u}, {t}, {w}, {z}, x, y ( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ (wyu uxt))
9190exbii 1582 . . . . . . . . . . . 12 (u{u}, {t}, {w}, {z}, x, y ( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) ↔ u(wyu uxt))
9230, 91bitri 240 . . . . . . . . . . 11 ({t}, {w}, {z}, x, y (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c) ↔ u(wyu uxt))
9329, 92anbi12i 678 . . . . . . . . . 10 (({t}, {w}, {z}, x, y Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) {t}, {w}, {z}, x, y (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) ↔ (z = w, t u(wyu uxt)))
947, 93bitri 240 . . . . . . . . 9 ({t}, {w}, {z}, x, y ( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) ↔ (z = w, t u(wyu uxt)))
9594exbii 1582 . . . . . . . 8 (t{t}, {w}, {z}, x, y ( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) ↔ t(z = w, t u(wyu uxt)))
966, 95bitri 240 . . . . . . 7 ({w}, {z}, x, y (( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) ↔ t(z = w, t u(wyu uxt)))
9796exbii 1582 . . . . . 6 (w{w}, {z}, x, y (( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) ↔ wt(z = w, t u(wyu uxt)))
985, 97bitri 240 . . . . 5 ({z}, x, y ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c) ↔ wt(z = w, t u(wyu uxt)))
992, 4, 983bitr4ri 269 . . . 4 ({z}, x, y ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c) ↔ z (x y))
10099releqmpt2 5809 . . 3 (((V × V) × V) (( Ins2 S Ins3 ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c)) “ 1c)) = (x V, y V (x y))
1011, 100eqtr4i 2376 . 2 Compose = (((V × V) × V) (( Ins2 S Ins3 ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c)) “ 1c))
102 vvex 4109 . . 3 V V
103 1stex 4739 . . . . . . . . . . 11 1st V
104103cnvex 5102 . . . . . . . . . 10 1st V
105102, 104xpex 5115 . . . . . . . . 9 (V × 1st ) V
106 2ndex 5112 . . . . . . . . . . 11 2nd V
107106cnvex 5102 . . . . . . . . . 10 2nd V
108107ins2ex 5797 . . . . . . . . 9 Ins2 2nd V
109105, 108inex 4105 . . . . . . . 8 ((V × 1st ) ∩ Ins2 2nd ) V
110109si3ex 5806 . . . . . . 7 SI3 ((V × 1st ) ∩ Ins2 2nd ) V
111110ins4ex 5799 . . . . . 6 Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) V
112 swapex 4742 . . . . . . . . . . . . 13 Swap V
113112si3ex 5806 . . . . . . . . . . . 12 SI3 Swap V
114113ins4ex 5799 . . . . . . . . . . 11 Ins4 SI3 Swap V
115 ssetex 4744 . . . . . . . . . . . . . . 15 S V
116115ins2ex 5797 . . . . . . . . . . . . . 14 Ins2 S V
117116ins2ex 5797 . . . . . . . . . . . . 13 Ins2 Ins2 S V
118117ins2ex 5797 . . . . . . . . . . . 12 Ins2 Ins2 Ins2 S V
119118ins2ex 5797 . . . . . . . . . . 11 Ins2 Ins2 Ins2 Ins2 S V
120114, 119inex 4105 . . . . . . . . . 10 ( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) V
121 1cex 4142 . . . . . . . . . 10 1c V
122120, 121imaex 4747 . . . . . . . . 9 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) V
123122ins2ex 5797 . . . . . . . 8 Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) V
124 idex 5504 . . . . . . . . . . . 12 I V
125124si3ex 5806 . . . . . . . . . . 11 SI3 I V
126125ins4ex 5799 . . . . . . . . . 10 Ins4 SI3 I V
127115ins3ex 5798 . . . . . . . . . . . . . 14 Ins3 S V
128127ins2ex 5797 . . . . . . . . . . . . 13 Ins2 Ins3 S V
129128ins2ex 5797 . . . . . . . . . . . 12 Ins2 Ins2 Ins3 S V
130129ins2ex 5797 . . . . . . . . . . 11 Ins2 Ins2 Ins2 Ins3 S V
131130ins2ex 5797 . . . . . . . . . 10 Ins2 Ins2 Ins2 Ins2 Ins3 S V
132126, 131inex 4105 . . . . . . . . 9 ( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) V
133132, 121imaex 4747 . . . . . . . 8 (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c) V
134123, 133inex 4105 . . . . . . 7 ( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) V
135134, 121imaex 4747 . . . . . 6 (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c) V
136111, 135inex 4105 . . . . 5 ( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) V
137136, 121imaex 4747 . . . 4 (( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) V
138137, 121imaex 4747 . . 3 ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c) V
139102, 102, 138mpt2exlem 5811 . 2 (((V × V) × V) (( Ins2 S Ins3 ((( Ins4 SI3 ((V × 1st ) ∩ Ins2 2nd ) ∩ (( Ins2 (( Ins4 SI3 Swap Ins2 Ins2 Ins2 Ins2 S ) “ 1c) ∩ (( Ins4 SI3 I ∩ Ins2 Ins2 Ins2 Ins2 Ins3 S ) “ 1c)) “ 1c)) “ 1c) “ 1c)) “ 1c)) V
140101, 139eqeltri 2423 1 Compose V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∩ cin 3208   ⊕ csymdif 3209  {csn 3737  1cc1c 4134  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   Swap cswap 4718   S csset 4719   ∘ ccom 4721   “ cima 4722   I cid 4763   × cxp 4770  ◡ccnv 4771  2nd c2nd 4783   ↦ cmpt2 5653   Compose ccompose 5747   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757 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-xp 4784  df-cnv 4785  df-2nd 4797  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758 This theorem is referenced by:  enmap2lem1  6063  enmap1lem1  6069
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