Theorem dmeq 2531

Description: Equality theorem for domain.

Assertion

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

Proof of Theorem dmeq

Step	Hyp	Ref	Expression
1		dmss 2530	. . 3 |- (A (_ B -> dom A (_ dom B)
2		dmss 2530	. . 3 |- (B (_ A -> dom B (_ dom A)
3	1, 2	anim12i 268	. 2 |- ((A (_ B /\ B (_ A) -> (dom A (_ dom B /\ dom B (_ dom A))
4		eqss 1516	. 2 |- (A = B <-> (A (_ B /\ B (_ A))
5		eqss 1516	. 2 |- (dom A = dom B <-> (dom A (_ dom B /\ dom B (_ dom A))
6	3, 4, 5	3imtr4 192	1 |- (A = B -> dom A = dom B)

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   (_ wss 1487  dom cdm 2410

This theorem is referenced by:  dmeqi 2532  dmeqd 2533  dmsnop 2547  fneq1 2718  cleqfv 2880  tfrlem10 2958  tz7.44lem1 2965  tz7.44-2 2967  tz7.44-3 2968  rdglem2 2976  aceq3 3556  ac7g 3570  infxpidmlem4 4936

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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