Theorem dmeq 4462

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 4461	. . 3 ⊢ (A ⊆ B → dom A ⊆ dom B)
2		dmss 4461	. . 3 ⊢ (B ⊆ A → dom B ⊆ dom A)
3	1, 2	anim12i 321	. 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (dom A ⊆ dom B ∧ dom B ⊆ dom A))
4		eqss 2937	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 2937	. 2 ⊢ (dom A = dom B ↔ (dom A ⊆ dom B ∧ dom B ⊆ dom A))
6	3, 4, 5	3imtr4i 190	1 ⊢ (A = B → dom A = dom B)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ⊆ wss 2894  dom cdm 4272

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-dm 4282

This theorem is referenced by:  dmeqi  4463  dmeqd  4464  xpid11m  4484  fneq1  4913  eqfnfv2  5191  offval  5642  ofrfval  5643  offval3  5684  smoeq  5827  tfrlemi14d  5868  tfrlemi14  5869  rdgi0g  5886  rdgivallem  5888  rdg0  5895  frec0g  5902  frecsuclem3  5906  frecsuc  5907  ereq1  6024