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Theorem dmco 5546
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 5225 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 5218 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 5260 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 5545 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 5225 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 5370 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2635 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2636 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  ccnv 5027  dom cdm 5028  ran crn 5029  cima 5031  ccom 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-xp 5034  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041
This theorem is referenced by:  curry1  7134  curry2  7137  smobeth  9265  hashkf  12939  imasless  15972  ofco2  20024  fcoinver  28592  xppreima  28623  smatrcl  28984  fco3  38210
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