Theorem dmco 6248

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 5880	. 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵)
2		cnvco 5870	. . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
3	2	rneqi 5922	. 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴)
4		rnco2 6247	. . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴)
5		dfdm4 5880	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
6	5	imaeq2i 6050	. . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴)
7	4, 6	eqtr4i 2762	. 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴)
8	1, 3, 7	3eqtri 2763	1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)

Syntax hints:   = wceq 1540  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662   ∘ ccom 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is referenced by:  fncofn  6660  curry1  8108  curry2  8111  smobeth  10605  hashkf  14355  imasless  17559  ofco2  22394  fcoinver  32590  xppreima  32628  smatrcl  33832