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Theorem dmco 5515
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 5195 . 2  |-  dom  ( A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 5188 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 5229 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5514 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `' A )
5 dfdm4 5195 . . . 4  |-  dom  A  =  ran  `' A
65imaeq2i 5335 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `' A )
74, 6eqtr4i 2499 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2500 1  |-  dom  ( A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  curry1  6876  curry2  6879  smobeth  8962  hashkf  12376  imasless  14798  ofco2  18760  fcoinver  27230  xppreima  27256
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