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Theorem rnco 4770
Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
rnco ran (AB) = ran (A ↾ ran B)

Proof of Theorem rnco
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 x V
2 vex 2554 . . . . . 6 y V
31, 2brco 4449 . . . . 5 (x(AB)yz(xBz zAy))
43exbii 1493 . . . 4 (x x(AB)yxz(xBz zAy))
5 excom 1551 . . . 4 (xz(xBz zAy) ↔ zx(xBz zAy))
6 ancom 253 . . . . . . 7 ((x xBz zAy) ↔ (zAy x xBz))
7 19.41v 1779 . . . . . . 7 (x(xBz zAy) ↔ (x xBz zAy))
8 vex 2554 . . . . . . . . 9 z V
98elrn 4520 . . . . . . . 8 (z ran Bx xBz)
109anbi2i 430 . . . . . . 7 ((zAy z ran B) ↔ (zAy x xBz))
116, 7, 103bitr4i 201 . . . . . 6 (x(xBz zAy) ↔ (zAy z ran B))
122brres 4561 . . . . . 6 (z(A ↾ ran B)y ↔ (zAy z ran B))
1311, 12bitr4i 176 . . . . 5 (x(xBz zAy) ↔ z(A ↾ ran B)y)
1413exbii 1493 . . . 4 (zx(xBz zAy) ↔ z z(A ↾ ran B)y)
154, 5, 143bitri 195 . . 3 (x x(AB)yz z(A ↾ ran B)y)
162elrn 4520 . . 3 (y ran (AB) ↔ x x(AB)y)
172elrn 4520 . . 3 (y ran (A ↾ ran B) ↔ z z(A ↾ ran B)y)
1815, 16, 173bitr4i 201 . 2 (y ran (AB) ↔ y ran (A ↾ ran B))
1918eqriv 2034 1 ran (AB) = ran (A ↾ ran B)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390   class class class wbr 3755  ran crn 4289  cres 4290  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by:  rnco2  4771  cofunexg  5680  1stcof  5732  2ndcof  5733
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