Theorem rneqi 4960

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 4959	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1397  ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  rnmpt  4980  resima  5046  resima2  5047  mptima  5088  ima0  5095  rnuni  5148  imaundi  5149  imaundir  5150  inimass  5153  dminxp  5181  imainrect  5182  xpima1  5183  xpima2m  5184  rnresv  5196  imacnvcnv  5201  rnpropg  5216  imadmres  5229  mptpreima  5230  dmco  5245  resdif  5605  fpr  5835  fprg  5836  fliftfuns  5938  rnoprab  6103  rnmpo  6131  qliftfuns  6787  xpassen  7013  sbthlemi6  7160  ennnfonelemrn  13039  cnconst2  14956  elply2  15458  iedgedgg  15911  edgiedgbg  15915  edg0iedg0g  15916  uhgrvtxedgiedgb  15993  uspgrf1oedg  16026  usgrf1oedg  16055  usgredg3  16064  ushgredgedg  16076  ushgredgedgloop  16078  0grsubgr  16114  edginwlkd  16205