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Theorem rnco 4754
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 2538 . . . . . 6 x V
2 vex 2538 . . . . . 6 y V
31, 2brco 4433 . . . . 5 (x(AB)yz(xBz zAy))
43exbii 1478 . . . 4 (x x(AB)yxz(xBz zAy))
5 excom 1536 . . . 4 (xz(xBz zAy) ↔ zx(xBz zAy))
6 ancom 253 . . . . . . 7 ((x xBz zAy) ↔ (zAy x xBz))
7 19.41v 1764 . . . . . . 7 (x(xBz zAy) ↔ (x xBz zAy))
8 vex 2538 . . . . . . . . 9 z V
98elrn 4504 . . . . . . . 8 (z ran Bx xBz)
109anbi2i 433 . . . . . . 7 ((zAy z ran B) ↔ (zAy x xBz))
116, 7, 103bitr4i 201 . . . . . 6 (x(xBz zAy) ↔ (zAy z ran B))
122brres 4545 . . . . . 6 (z(A ↾ ran B)y ↔ (zAy z ran B))
1311, 12bitr4i 176 . . . . 5 (x(xBz zAy) ↔ z(A ↾ ran B)y)
1413exbii 1478 . . . 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 4504 . . 3 (y ran (AB) ↔ x x(AB)y)
172elrn 4504 . . 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 2019 1 ran (AB) = ran (A ↾ ran B)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374   class class class wbr 3738  ran crn 4273  cres 4274  ccom 4276
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284
This theorem is referenced by:  rnco2  4755  cofunexg  5661  1stcof  5713  2ndcof  5714
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