Theorem reseq1i 5057

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 5055	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1402   ↾ cres 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-res 4784

This theorem is referenced by:  reseq12i  5059  resindm  5103  resmpt  5109  resmpt3  5110  resmptf  5111  opabresid  5114  rescnvcnv  5248  coires1  5303  fresaunres1disj  5569  fcoi1  5570  fvsnun1  5906  fvsnun2  5907  resoprab  6178  resmpo  6180  ofmres  6363  f1stres  6387  f2ndres  6388  df1st2  6449  df2nd2  6450  dftpos2  6526  tfr2a  6586  freccllem  6667  frecfcllem  6669  frecsuclem  6671  djuf1olemr  7388  divfnzn  10004  gsummptfidmadd  14144  cnmptid  15365  xmsxmet2  15547  msmet2  15548  cnfldms  15620