Theorem rneqi 4958

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 4957	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1395  ran crn 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  rnmpt  4978  resima  5044  resima2  5045  mptima  5086  ima0  5093  rnuni  5146  imaundi  5147  imaundir  5148  inimass  5151  dminxp  5179  imainrect  5180  xpima1  5181  xpima2m  5182  rnresv  5194  imacnvcnv  5199  rnpropg  5214  imadmres  5227  mptpreima  5228  dmco  5243  resdif  5602  fpr  5831  fprg  5832  fliftfuns  5934  rnoprab  6099  rnmpo  6127  qliftfuns  6783  xpassen  7009  sbthlemi6  7152  ennnfonelemrn  13030  cnconst2  14947  elply2  15449  iedgedgg  15902  edgiedgbg  15906  edg0iedg0g  15907  uhgrvtxedgiedgb  15982  uspgrf1oedg  16015  usgrf1oedg  16044  usgredg3  16053  ushgredgedg  16065  ushgredgedgloop  16067  edginwlkd  16152