Theorem rneq 4504

Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Assertion

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

Proof of Theorem rneq

Step	Hyp	Ref	Expression
1		cnveq 4452	. . 3 ⊢ (A = B → ◡A = ◡B)
2	1	dmeqd 4480	. 2 ⊢ (A = B → dom ◡A = dom ◡B)
3		df-rn 4299	. 2 ⊢ ran A = dom ◡A
4		df-rn 4299	. 2 ⊢ ran B = dom ◡B
5	2, 3, 4	3eqtr4g 2094	1 ⊢ (A = B → ran A = ran B)

Syntax hints:   → wi 4   = wceq 1242  ^◡ccnv 4287  dom cdm 4288  ran crn 4289

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-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  rneqi  4505  rneqd  4506  xpima1  4710  feq1  4973  foeq1  5045