Theorem rneqi 4807

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 4806	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1332  ran crn 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-cnv 4587  df-dm 4589  df-rn 4590

This theorem is referenced by:  rnmpt  4827  resima  4892  resima2  4893  ima0  4938  rnuni  4990  imaundi  4991  imaundir  4992  inimass  4995  dminxp  5023  imainrect  5024  xpima1  5025  xpima2m  5026  rnresv  5038  imacnvcnv  5043  rnpropg  5058  imadmres  5071  mptpreima  5072  dmco  5087  resdif  5429  fpr  5642  fprg  5643  fliftfuns  5739  rnoprab  5894  rnmpo  5921  qliftfuns  6553  xpassen  6764  sbthlemi6  6895  ennnfonelemrn  12099  cnconst2  12572