Theorem reseq1i 4942

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 4940	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

Syntax hints:   = wceq 1364   ↾ cres 4665

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-res 4675

This theorem is referenced by:  reseq12i  4944  resindm  4988  resmpt  4994  resmpt3  4995  resmptf  4996  opabresid  4999  rescnvcnv  5132  coires1  5187  fcoi1  5438  fvsnun1  5759  fvsnun2  5760  resoprab  6018  resmpo  6020  ofmres  6193  f1stres  6217  f2ndres  6218  df1st2  6277  df2nd2  6278  dftpos2  6319  tfr2a  6379  freccllem  6460  frecfcllem  6462  frecsuclem  6464  djuf1olemr  7120  divfnzn  9695  cnmptid  14517  xmsxmet2  14699  msmet2  14700  cnfldms  14772