Theorem rneqd 4677

Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.)

Hypothesis

Ref	Expression
rneqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
rneqd	⊢ (𝜑 → ran 𝐴 = ran 𝐵)

Proof of Theorem rneqd

Step	Hyp	Ref	Expression
1		rneqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		rneq 4675	. 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → ran 𝐴 = ran 𝐵)

Syntax hints:   → wi 4   = wceq 1290  ran crn 4453

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463

This theorem is referenced by:  resima2  4759  imaeq1  4782  imaeq2  4783  resiima  4803  elxp4  4931  elxp5  4932  funimacnv  5103  funimaexg  5111  fnima  5145  fnrnfv  5364  2ndvalg  5928  fo2nd  5943  f2ndres  5945  en1  6570  xpassen  6600  xpdom2  6601  sbthlemi4  6723  djudom  6837  exmidfodomrlemim  6888  iseqeq1  9919  iseqeq2  9920  iseqeq3  9921  iseqval  9932  iseqvalt  9934  seq3val  9935  restval  11719  restid2  11722