Theorem rneqi 4920

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 4919	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1373  ran crn 4689

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4055  df-opab 4117  df-cnv 4696  df-dm 4698  df-rn 4699

This theorem is referenced by:  rnmpt  4940  resima  5006  resima2  5007  mptima  5048  ima0  5055  rnuni  5108  imaundi  5109  imaundir  5110  inimass  5113  dminxp  5141  imainrect  5142  xpima1  5143  xpima2m  5144  rnresv  5156  imacnvcnv  5161  rnpropg  5176  imadmres  5189  mptpreima  5190  dmco  5205  resdif  5561  fpr  5784  fprg  5785  fliftfuns  5885  rnoprab  6046  rnmpo  6074  qliftfuns  6724  xpassen  6945  sbthlemi6  7085  ennnfonelemrn  12875  cnconst2  14790  elply2  15292  iedgedgg  15742  edgiedgbg  15746  edg0iedg0g  15747  uhgrvtxedgiedgb  15817