Theorem dmeq 4553

Description: Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)

Assertion

Ref	Expression
dmeq	|- ( A = B -> dom A = dom B )

Proof of Theorem dmeq

Step	Hyp	Ref	Expression
1		dmss 4552	. . 3 |- ( A C_ B -> dom A C_ dom B )
2		dmss 4552	. . 3 |- ( B C_ A -> dom B C_ dom A )
3	1, 2	anim12i 331	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( dom A C_ dom B /\ dom B C_ dom A ) )
4		eqss 3014	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 3014	. 2 |- ( dom A = dom B <-> ( dom A C_ dom B /\ dom B C_ dom A ) )
6	3, 4, 5	3imtr4i 199	1 |- ( A = B -> dom A = dom B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   dom cdm 4363

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-dm 4373

This theorem is referenced by:  dmeqi  4554  dmeqd  4555  xpid11m  4575  fneq1  5007  eqfnfv2  5287  offval  5739  ofrfval  5740  offval3  5781  smoeq  5928  tfrlemi14d  5970  rdgivallem  5991  rdg0  5997  frec0g  6006  frecsuclem3  6013  frecsuc  6014  ereq1  6136  fundmeng  6310