Theorem dmcoeq 5922

Description: Domain of a composition. (Contributed by NM, 19-Mar-1998.)

Assertion

Ref	Expression
dmcoeq	⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Proof of Theorem dmcoeq

Step	Hyp	Ref	Expression
1		eqimss2 3995	. 2 ⊢ (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2		dmcosseq 5919	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)
3	1, 2	syl 17	1 ⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3903  dom cdm 5619  ran crn 5620   ∘ ccom 5623

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-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630

This theorem is referenced by:  rncoeq  5923  dfdm2  6229  funcocnv2  6789