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Theorem dmco 5154
Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
Assertion
Ref Expression
dmco  |-  dom  (  A  o.  B )  =  ( `' B " dom  A )

Proof of Theorem dmco
StepHypRef Expression
1 dfdm4 4846 . 2  |-  dom  (  A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 4839 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 4879 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5153 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `'  A )
5 dfdm4 4846 . . . 4  |-  dom  A  =  ran  `'  A
65imaeq2i 4984 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `'  A )
74, 6eqtr4i 2279 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2280 1  |-  dom  (  A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   `'ccnv 4646   dom cdm 4647   ran crn 4648   "cima 4650    o. ccom 4651
This theorem is referenced by:  curry1  6130  curry2  6133  smobeth  8162  hashkf  11291  imasless  13390  domrancur1b  24553  domrancur1c  24555
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668
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