Theorem dmeq 4535

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 4534	. . 3 |- ( A C_ B -> dom A C_ dom B )
2		dmss 4534	. . 3 |- ( B C_ A -> dom B C_ dom A )
3	1, 2	anim12i 321	. 2 |- ( ( A C_ B /\ B C_ A ) -> ( dom A C_ dom B /\ dom B C_ dom A ) )
4		eqss 2960	. 2 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
5		eqss 2960	. 2 |- ( dom A = dom B <-> ( dom A C_ dom B /\ dom B C_ dom A ) )
6	3, 4, 5	3imtr4i 190	1 |- ( A = B -> dom A = dom B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   C_ wss 2917   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by:  dmeqi  4536  dmeqd  4537  xpid11m  4557  fneq1  4987  eqfnfv2  5266  offval  5719  ofrfval  5720  offval3  5761  smoeq  5905  tfrlemi14d  5947  rdgivallem  5968  rdg0  5974  frec0g  5983  frecsuclem3  5990  frecsuc  5991  ereq1  6113  fundmeng  6287