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Theorem dmco 5337
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 5022 . 2  |-  dom  ( A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 5015 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 5055 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5336 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `' A )
5 dfdm4 5022 . . . 4  |-  dom  A  =  ran  `' A
65imaeq2i 5160 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `' A )
74, 6eqtr4i 2427 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2428 1  |-  dom  ( A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840    o. ccom 4841
This theorem is referenced by:  curry1  6397  curry2  6400  smobeth  8417  hashkf  11575  imasless  13720  xppreima  24012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850
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