Theorem dmcoeq 5968

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 3972	. 2 ⊢ (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2		dmcosseq 5966	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)
3	1, 2	syl 18	1 ⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900  dom cdm 5655  ran crn 5656   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666

This theorem is used by:  dmcocnv1  5969  dmcocnv2  5970  rncoeq  5971  funcocnv2OLD  6870