Theorem rneqi 4905

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 4904	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1372  ran crn 4675

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4682  df-dm 4684  df-rn 4685

This theorem is referenced by:  rnmpt  4925  resima  4991  resima2  4992  mptima  5033  ima0  5040  rnuni  5093  imaundi  5094  imaundir  5095  inimass  5098  dminxp  5126  imainrect  5127  xpima1  5128  xpima2m  5129  rnresv  5141  imacnvcnv  5146  rnpropg  5161  imadmres  5174  mptpreima  5175  dmco  5190  resdif  5543  fpr  5765  fprg  5766  fliftfuns  5866  rnoprab  6027  rnmpo  6055  qliftfuns  6705  xpassen  6924  sbthlemi6  7063  ennnfonelemrn  12732  cnconst2  14647  elply2  15149