Theorem reseq1i 5015

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 5013	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1398   ↾ cres 4733

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-res 4743

This theorem is referenced by:  reseq12i  5017  resindm  5061  resmpt  5067  resmpt3  5068  resmptf  5069  opabresid  5072  rescnvcnv  5206  coires1  5261  fcoi1  5525  fvsnun1  5859  fvsnun2  5860  resoprab  6127  resmpo  6129  ofmres  6307  f1stres  6331  f2ndres  6332  df1st2  6393  df2nd2  6394  dftpos2  6470  tfr2a  6530  freccllem  6611  frecfcllem  6613  frecsuclem  6615  djuf1olemr  7296  divfnzn  9898  cnmptid  15072  xmsxmet2  15254  msmet2  15255  cnfldms  15327