Theorem dmco 6227

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 5859	. 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵)
2		cnvco 5849	. . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
3	2	rneqi 5901	. 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴)
4		rnco2 6226	. . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴)
5		dfdm4 5859	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
6	5	imaeq2i 6029	. . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴)
7	4, 6	eqtr4i 2755	. 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴)
8	1, 3, 7	3eqtri 2756	1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)

Syntax hints:   = wceq 1540  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   “ cima 5641   ∘ ccom 5642

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651

This theorem is referenced by:  fncofn  6635  curry1  8083  curry2  8086  smobeth  10539  hashkf  14297  imasless  17503  ofco2  22338  fcoinver  32533  xppreima  32569  smatrcl  33786