Theorem dmcoeq 4705

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 3079	. 2 ⊢ (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2		dmcosseq 4704	. 2 ⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)
3	1, 2	syl 14	1 ⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

Syntax hints:   → wi 4   = wceq 1289   ⊆ wss 2999  dom cdm 4438  ran crn 4439   ∘ ccom 4442

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449

This theorem is referenced by:  rncoeq  4706  dfdm2  4965  funcocnv2  5278