Theorem reseq1i 5039

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 5037	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1398   ↾ cres 4756

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-res 4766

This theorem is referenced by:  reseq12i  5041  resindm  5085  resmpt  5091  resmpt3  5092  resmptf  5093  opabresid  5096  rescnvcnv  5230  coires1  5285  fresaunres1disj  5551  fcoi1  5552  fvsnun1  5886  fvsnun2  5887  resoprab  6157  resmpo  6159  ofmres  6342  f1stres  6366  f2ndres  6367  df1st2  6428  df2nd2  6429  dftpos2  6505  tfr2a  6565  freccllem  6646  frecfcllem  6648  frecsuclem  6650  djuf1olemr  7358  divfnzn  9971  cnmptid  15258  xmsxmet2  15440  msmet2  15441  cnfldms  15513