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Theorem dmco 6104
Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
Assertion
Ref Expression
dmco dom (𝐴𝐵) = (𝐵 “ dom 𝐴)

Proof of Theorem dmco
StepHypRef Expression
1 dfdm4 5762 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 5754 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 5805 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 6103 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 5762 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 5924 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2851 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2852 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  ccnv 5552  dom cdm 5553  ran crn 5554  cima 5556  ccom 5557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566
This theorem is referenced by:  curry1  7793  curry2  7796  smobeth  10000  hashkf  13685  imasless  16805  ofco2  20978  fcoinver  30274  xppreima  30311  smatrcl  30949  fco3  41358
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