Theorem reseq1i 4997

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 4995	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1395   ↾ cres 4718

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 4728

This theorem is referenced by:  reseq12i  4999  resindm  5043  resmpt  5049  resmpt3  5050  resmptf  5051  opabresid  5054  rescnvcnv  5187  coires1  5242  fcoi1  5502  fvsnun1  5829  fvsnun2  5830  resoprab  6091  resmpo  6093  ofmres  6271  f1stres  6295  f2ndres  6296  df1st2  6355  df2nd2  6356  dftpos2  6397  tfr2a  6457  freccllem  6538  frecfcllem  6540  frecsuclem  6542  djuf1olemr  7209  divfnzn  9804  cnmptid  14940  xmsxmet2  15122  msmet2  15123  cnfldms  15195