Theorem rneqi 5005

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 5004	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1402  ran crn 4770

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 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 theorem 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 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  rnmpt  5025  resima  5091  resima2  5092  mptima  5133  ima0  5141  rnuni  5194  imaundi  5195  imaundir  5196  inimass  5199  dminxp  5227  imainrect  5228  xpima1  5229  xpima2m  5230  rnresv  5242  imacnvcnv  5247  rnpropg  5262  imadmres  5275  mptpreima  5276  dmco  5291  resdif  5656  fpr  5888  fprg  5889  fliftfuns  5994  rnoprab  6161  rnmpo  6189  qliftfuns  6883  xpassen  7118  sbthlemi6  7269  ennnfonelemrn  13288  cnconst2  15257  elply2  15759  iedgedgg  16216  edgiedgbg  16220  edg0iedg0g  16221  uhgrvtxedgiedgb  16298  uspgrf1oedg  16331  usgrf1oedg  16360  usgredg3  16369  ushgredgedg  16381  ushgredgedgloop  16383  0grsubgr  16419  edginwlkd  16510