Theorem rneqi 4890

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 4889	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1364  ran crn 4660

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  rnmpt  4910  resima  4975  resima2  4976  mptima  5017  ima0  5024  rnuni  5077  imaundi  5078  imaundir  5079  inimass  5082  dminxp  5110  imainrect  5111  xpima1  5112  xpima2m  5113  rnresv  5125  imacnvcnv  5130  rnpropg  5145  imadmres  5158  mptpreima  5159  dmco  5174  resdif  5522  fpr  5740  fprg  5741  fliftfuns  5841  rnoprab  6001  rnmpo  6029  qliftfuns  6673  xpassen  6884  sbthlemi6  7021  ennnfonelemrn  12576  cnconst2  14401  elply2  14881