Theorem rneqi 4856

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 4855	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1353  ran crn 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-cnv 4635  df-dm 4637  df-rn 4638

This theorem is referenced by:  rnmpt  4876  resima  4941  resima2  4942  ima0  4988  rnuni  5041  imaundi  5042  imaundir  5043  inimass  5046  dminxp  5074  imainrect  5075  xpima1  5076  xpima2m  5077  rnresv  5089  imacnvcnv  5094  rnpropg  5109  imadmres  5122  mptpreima  5123  dmco  5138  resdif  5484  fpr  5699  fprg  5700  fliftfuns  5799  rnoprab  5958  rnmpo  5985  qliftfuns  6619  xpassen  6830  sbthlemi6  6961  ennnfonelemrn  12420  cnconst2  13736