Theorem reseq1i 5001

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 4999	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1395   ↾ cres 4721

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 4731

This theorem is referenced by:  reseq12i  5003  resindm  5047  resmpt  5053  resmpt3  5054  resmptf  5055  opabresid  5058  rescnvcnv  5191  coires1  5246  fcoi1  5508  fvsnun1  5840  fvsnun2  5841  resoprab  6106  resmpo  6108  ofmres  6287  f1stres  6311  f2ndres  6312  df1st2  6371  df2nd2  6372  dftpos2  6413  tfr2a  6473  freccllem  6554  frecfcllem  6556  frecsuclem  6558  djuf1olemr  7229  divfnzn  9824  cnmptid  14963  xmsxmet2  15145  msmet2  15146  cnfldms  15218