Theorem rneq 2555

Description: Equality theorem for range.

Assertion

Ref	Expression
rneq	⊢ (A = B → ran A = ran B)

Proof of Theorem rneq

Step	Hyp	Ref	Expression
1		cnveq 2513	. . 3 ⊢ (A = B → ◡A = ◡B)
2	1	dmeqd 2533	. 2 ⊢ (A = B → dom ◡A = dom ◡B)
3		df-rn 2429	. 2 ⊢ ran A = dom ◡A
4		df-rn 2429	. 2 ⊢ ran B = dom ◡B
5	2, 3, 4	3eqtr4g 1147	1 ⊢ (A = B → ran A = ran B)

Syntax hints:   → wi 2   = wceq 1091  ^◡ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  rneqi 2556  rneqd 2557  feq1 2748  foeq1 2784  fvres 2840  tz7.44-3 2968  rdglem2 2976  map0e 3266  aceq5lem3 3560  numthlem 3598  numth 3599  zornlem1 3603  zorn 3611  infxpidmlem4 4936  infxpidmlem8 4940  infxpidmlem10 4942  infmap2lem2 4952  pj11 5591  pjss1co 5633

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

metamath.org