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Theorem dmco 5087
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 4779 . 2  |-  dom  (  A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 4772 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 4812 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5086 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `'  A )
5 dfdm4 4779 . . . 4  |-  dom  A  =  ran  `'  A
65imaeq2i 4917 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `'  A )
74, 6eqtr4i 2276 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2277 1  |-  dom  (  A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   `'ccnv 4579   dom cdm 4580   ran crn 4581   "cima 4583    o. ccom 4584
This theorem is referenced by:  curry1  6062  curry2  6065  smobeth  8088  hashkf  11217  imasless  13316  domrancur1b  24366  domrancur1c  24368
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601
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