Theorem dmco 6209

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 5841	. 2 ⊢ dom (𝐴 ∘ 𝐵) = ran ◡(𝐴 ∘ 𝐵)
2		cnvco 5831	. . 3 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
3	2	rneqi 5883	. 2 ⊢ ran ◡(𝐴 ∘ 𝐵) = ran (◡𝐵 ∘ ◡𝐴)
4		rnco2 6208	. . 3 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ ran ◡𝐴)
5		dfdm4 5841	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
6	5	imaeq2i 6013	. . 3 ⊢ (◡𝐵 “ dom 𝐴) = (◡𝐵 “ ran ◡𝐴)
7	4, 6	eqtr4i 2759	. 2 ⊢ ran (◡𝐵 ∘ ◡𝐴) = (◡𝐵 “ dom 𝐴)
8	1, 3, 7	3eqtri 2760	1 ⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)

Syntax hints:   = wceq 1541  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   “ cima 5624   ∘ ccom 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by:  fncofn  6605  curry1  8042  curry2  8045  smobeth  10486  hashkf  14243  imasless  17448  ofco2  22369  fcoinver  32588  xppreima  32631  smatrcl  33832