Theorem rneqi 4952

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 4951	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1395  ran crn 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  rnmpt  4972  resima  5038  resima2  5039  mptima  5080  ima0  5087  rnuni  5140  imaundi  5141  imaundir  5142  inimass  5145  dminxp  5173  imainrect  5174  xpima1  5175  xpima2m  5176  rnresv  5188  imacnvcnv  5193  rnpropg  5208  imadmres  5221  mptpreima  5222  dmco  5237  resdif  5596  fpr  5825  fprg  5826  fliftfuns  5928  rnoprab  6093  rnmpo  6121  qliftfuns  6774  xpassen  6997  sbthlemi6  7140  ennnfonelemrn  13005  cnconst2  14922  elply2  15424  iedgedgg  15876  edgiedgbg  15880  edg0iedg0g  15881  uhgrvtxedgiedgb  15956  uspgrf1oedg  15989  usgrf1oedg  16018  usgredg3  16027  ushgredgedg  16039  ushgredgedgloop  16041  edginwlkd  16096