Theorem dmco 6088

Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)

Assertion

Ref	Expression
dmco	⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)

Proof of Theorem dmco

Step	Hyp	Ref	Expression
1		dfdm4 5740	. 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵)
2		cnvco 5730	. . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
3	2	rneqi 5782	. 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴)
4		rnco2 6087	. . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴)
5		dfdm4 5740	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
6	5	imaeq2i 5903	. . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴)
7	4, 6	eqtr4i 2784	. 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴)
8	1, 3, 7	3eqtri 2785	1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)

Syntax hints:   = wceq 1538  ^◡ccnv 5526  dom cdm 5527  ran crn 5528   “ cima 5530   ∘ ccom 5531

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540

This theorem is referenced by:  curry1  7809  curry2  7812  smobeth  10051  hashkf  13747  imasless  16876  ofco2  21156  fcoinver  30473  xppreima  30510  smatrcl  31271  fco3  42253