Theorem dmeq 4478

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 4477	. . 3 ⊢ (A ⊆ B → dom A ⊆ dom B)
2		dmss 4477	. . 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 2954	. 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
5		eqss 2954	. 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 1242   ⊆ wss 2911  dom cdm 4288

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298

This theorem is referenced by:  dmeqi  4479  dmeqd  4480  xpid11m  4500  fneq1  4930  eqfnfv2  5209  offval  5661  ofrfval  5662  offval3  5703  smoeq  5846  tfrlemi14d  5888  rdgivallem  5908  rdg0  5914  frec0g  5922  frecsuclem3  5929  frecsuc  5930  ereq1  6049  fundmeng  6223