Theorem rneqi 4853

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 4852	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	ax-mp 5	1 ⊢ ran 𝐴 = ran 𝐵

Syntax hints:   = wceq 1353  ran crn 4626

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 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-cnv 4633  df-dm 4635  df-rn 4636

This theorem is referenced by:  rnmpt  4873  resima  4938  resima2  4939  ima0  4985  rnuni  5038  imaundi  5039  imaundir  5040  inimass  5043  dminxp  5071  imainrect  5072  xpima1  5073  xpima2m  5074  rnresv  5086  imacnvcnv  5091  rnpropg  5106  imadmres  5119  mptpreima  5120  dmco  5135  resdif  5481  fpr  5696  fprg  5697  fliftfuns  5795  rnoprab  5954  rnmpo  5981  qliftfuns  6615  xpassen  6826  sbthlemi6  6957  ennnfonelemrn  12411  cnconst2  13595