Theorem rneqi 5012

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 5011	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	ax-mp 5	1 |- ran A = ran B

This proof depends on syntax axioms:   = wceq 1402   ran crn 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-cnv 4784  df-dm 4786  df-rn 4787

This theorem is used by:  rnmpt  5032  resima  5098  resima2  5099  mptima  5141  ima0  5149  rnuni  5202  imaundi  5203  imaundir  5204  inimass  5207  dminxp  5235  imainrect  5236  xpima1  5237  xpima2m  5238  rnresv  5250  imacnvcnv  5255  rnpropg  5270  imadmres  5283  mptpreima  5284  dmco  5299  resdif  5664  fpr  5900  fprg  5901  fliftfuns  6007  rnoprab  6174  rnmpo  6202  qliftfuns  6898  xpassen  7134  sbthlemi6  7286  ennnfonelemrn  13367  cnconst2  15476  elply2  15978  iedgedgg  16524  edgiedgbg  16528  edg0iedg0g  16529  uhgrvtxedgiedgb  16606  uspgrf1oedg  16639  usgrf1oedg  16668  usgredg3  16677  ushgredgedg  16689  ushgredgedgloop  16691  0grsubgr  16727  edginwlkd  16818