Theorem rneqi 4951

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 4950	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1395  ran crn 4719

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 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729

This theorem is referenced by:  rnmpt  4971  resima  5037  resima2  5038  mptima  5079  ima0  5086  rnuni  5139  imaundi  5140  imaundir  5141  inimass  5144  dminxp  5172  imainrect  5173  xpima1  5174  xpima2m  5175  rnresv  5187  imacnvcnv  5192  rnpropg  5207  imadmres  5220  mptpreima  5221  dmco  5236  resdif  5593  fpr  5820  fprg  5821  fliftfuns  5921  rnoprab  6086  rnmpo  6114  qliftfuns  6764  xpassen  6985  sbthlemi6  7125  ennnfonelemrn  12985  cnconst2  14901  elply2  15403  iedgedgg  15855  edgiedgbg  15859  edg0iedg0g  15860  uhgrvtxedgiedgb  15935  uspgrf1oedg  15968  usgrf1oedg  15997  usgredg3  16006  ushgredgedg  16018  ushgredgedgloop  16020