Theorem reseq1i 4896

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 4894	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1353   ↾ cres 4622

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-res 4632

This theorem is referenced by:  reseq12i  4898  resindm  4942  resmpt  4948  resmpt3  4949  resmptf  4950  opabresid  4953  rescnvcnv  5083  coires1  5138  fcoi1  5388  fvsnun1  5705  fvsnun2  5706  resoprab  5961  resmpo  5963  ofmres  6127  f1stres  6150  f2ndres  6151  df1st2  6210  df2nd2  6211  dftpos2  6252  tfr2a  6312  freccllem  6393  frecfcllem  6395  frecsuclem  6397  djuf1olemr  7043  divfnzn  9594  cnmptid  13352  xmsxmet2  13534  msmet2  13535