Theorem rneqi 4966

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 4965	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1398  ran crn 4732

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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  rnmpt  4986  resima  5052  resima2  5053  mptima  5094  ima0  5102  rnuni  5155  imaundi  5156  imaundir  5157  inimass  5160  dminxp  5188  imainrect  5189  xpima1  5190  xpima2m  5191  rnresv  5203  imacnvcnv  5208  rnpropg  5223  imadmres  5236  mptpreima  5237  dmco  5252  resdif  5614  fpr  5844  fprg  5845  fliftfuns  5949  rnoprab  6114  rnmpo  6142  qliftfuns  6831  xpassen  7057  sbthlemi6  7204  ennnfonelemrn  13103  cnconst2  15027  elply2  15529  iedgedgg  15985  edgiedgbg  15989  edg0iedg0g  15990  uhgrvtxedgiedgb  16067  uspgrf1oedg  16100  usgrf1oedg  16129  usgredg3  16138  ushgredgedg  16150  ushgredgedgloop  16152  0grsubgr  16188  edginwlkd  16279