Theorem dmco 5174

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 4854	. 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵)
2		cnvco 4847	. . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
3	2	rneqi 4890	. 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴)
4		rnco2 5173	. . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴)
5		dfdm4 4854	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
6	5	imaeq2i 5003	. . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴)
7	4, 6	eqtr4i 2217	. 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴)
8	1, 3, 7	3eqtri 2218	1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)

Syntax hints:   = wceq 1364  ^◡ccnv 4658  dom cdm 4659  ran crn 4660   “ cima 4662   ∘ ccom 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  casedm  7145  caseinl  7150  caseinr  7151  djudm  7164