ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rncoss Unicode version

Theorem rncoss 4703
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoss  |-  ran  ( A  o.  B )  C_ 
ran  A

Proof of Theorem rncoss
StepHypRef Expression
1 dmcoss 4702 . 2  |-  dom  ( `' B  o.  `' A )  C_  dom  `' A
2 df-rn 4449 . . 3  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
3 cnvco 4621 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
43dmeqi 4637 . . 3  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
52, 4eqtri 2108 . 2  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
6 df-rn 4449 . 2  |-  ran  A  =  dom  `' A
71, 5, 63sstr4i 3065 1  |-  ran  ( A  o.  B )  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    C_ wss 2999   `'ccnv 4437   dom cdm 4438   ran crn 4439    o. ccom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449
This theorem is referenced by:  cossxp  4953  fco  5176  caseinj  6780  djuinj  6786
  Copyright terms: Public domain W3C validator