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Theorem dmco 5091
 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 4775 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 4768 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 4811 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 5090 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 4775 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 4923 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2181 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2182 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
 Colors of variables: wff set class Syntax hints:   = wceq 1335  ◡ccnv 4582  dom cdm 4583  ran crn 4584   “ cima 4586   ∘ ccom 4587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4589  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596 This theorem is referenced by:  casedm  7020  caseinl  7025  caseinr  7026  djudm  7039
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