MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmco Structured version   Visualization version   GIF version

Theorem dmco 6250
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 5893 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 5883 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 5934 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 6249 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 5893 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 6055 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2763 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2764 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  ccnv 5674  dom cdm 5675  ran crn 5676  cima 5678  ccom 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by:  fncofn  6663  fco3OLD  6748  curry1  8086  curry2  8089  smobeth  10577  hashkf  14288  imasless  17482  ofco2  21944  fcoinver  31822  xppreima  31858  smatrcl  32764
  Copyright terms: Public domain W3C validator