Theorem reseq1i 4855

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 4853	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1332   ↾ cres 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-in 3104  df-res 4591

This theorem is referenced by:  reseq12i  4857  resindm  4901  resmpt  4907  resmpt3  4908  resmptf  4909  opabresid  4912  rescnvcnv  5041  coires1  5096  fcoi1  5343  fvsnun1  5657  fvsnun2  5658  resoprab  5907  resmpo  5909  ofmres  6074  f1stres  6097  f2ndres  6098  df1st2  6156  df2nd2  6157  dftpos2  6198  tfr2a  6258  freccllem  6339  frecfcllem  6341  frecsuclem  6343  djuf1olemr  6984  divfnzn  9508  cnmptid  12628  xmsxmet2  12810  msmet2  12811