Theorem dmcoeq 3358

Description: Domain of a composition.

Assertion

Ref	Expression
dmcoeq	|- (dom A = ran B -> dom ( A o. B) = dom B)

Proof of Theorem dmcoeq

Step	Hyp	Ref	Expression
1		eqimss2 2106	. 2 |- (dom A = ran B -> ran B (_ dom A)
2		dmcosseq 3357	. 2 |- (ran B (_ dom A -> dom ( A o. B) = dom B)
3	1, 2	syl 10	1 |- (dom A = ran B -> dom ( A o. B) = dom B)

Syntax hints:   -> wi 3   = wceq 954   (_ wss 2043  dom cdm 3165  ran crn 3166   o. ccom 3169

This theorem is referenced by:  rncoeq 3359  f1ococnv2 3699  vsfval 8206

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184

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