Theorem dmcoeq 5883

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 3978	. 2 ⊢ (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2		dmcosseq 5882	. 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 1539   ⊆ wss 3887  dom cdm 5589  ran crn 5590   ∘ ccom 5593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600

This theorem is referenced by:  rncoeq  5884  dfdm2  6184  funcocnv2  6741