Theorem rneqi 4580

Description: Equality inference for range. (Contributed by NM, 4-Mar-2004.)

Hypothesis

Ref	Expression
rneqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rneqi	⊢ ran 𝐴 = ran 𝐵

Proof of Theorem rneqi

Step	Hyp	Ref	Expression
1		rneqi.1	. 2 ⊢ 𝐴 = 𝐵
2		rneq 4579	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 7	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1284  ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  rnmpt  4600  resima  4661  resima2  4662  ima0  4704  rnuni  4755  imaundi  4756  imaundir  4757  inimass  4760  dminxp  4785  imainrect  4786  xpima1  4787  xpima2m  4788  rnresv  4800  imacnvcnv  4805  rnpropg  4820  imadmres  4833  mptpreima  4834  dmco  4849  resdif  5168  fpr  5366  fprg  5367  fliftfuns  5458  rnoprab  5607  rnmpt2  5631  qliftfuns  6213  xpassen  6327