Theorem rneqi 4990

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 4989	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1398  ran crn 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  rnmpt  5010  resima  5076  resima2  5077  mptima  5118  ima0  5126  rnuni  5179  imaundi  5180  imaundir  5181  inimass  5184  dminxp  5212  imainrect  5213  xpima1  5214  xpima2m  5215  rnresv  5227  imacnvcnv  5232  rnpropg  5247  imadmres  5260  mptpreima  5261  dmco  5276  resdif  5641  fpr  5871  fprg  5872  fliftfuns  5977  rnoprab  6144  rnmpo  6172  qliftfuns  6866  xpassen  7094  sbthlemi6  7245  ennnfonelemrn  13254  cnconst2  15224  elply2  15726  iedgedgg  16182  edgiedgbg  16186  edg0iedg0g  16187  uhgrvtxedgiedgb  16264  uspgrf1oedg  16297  usgrf1oedg  16326  usgredg3  16335  ushgredgedg  16347  ushgredgedgloop  16349  0grsubgr  16385  edginwlkd  16476