Theorem reseq1i 5004

Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
reseq1i	⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Proof of Theorem reseq1i

Step	Hyp	Ref	Expression
1		reseqi.1	. 2 ⊢ 𝐴 = 𝐵
2		reseq1 5002	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1395   ↾ cres 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-res 4732

This theorem is referenced by:  reseq12i  5006  resindm  5050  resmpt  5056  resmpt3  5057  resmptf  5058  opabresid  5061  rescnvcnv  5194  coires1  5249  fcoi1  5511  fvsnun1  5843  fvsnun2  5844  resoprab  6109  resmpo  6111  ofmres  6290  f1stres  6314  f2ndres  6315  df1st2  6376  df2nd2  6377  dftpos2  6418  tfr2a  6478  freccllem  6559  frecfcllem  6561  frecsuclem  6563  djuf1olemr  7237  divfnzn  9833  cnmptid  14976  xmsxmet2  15158  msmet2  15159  cnfldms  15231