Theorem reseq1i 5061

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 5059	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)

This proof depends on syntax axioms:   = wceq 1402   ↾ cres 4778

This proof depends on 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 proof 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 4788

This theorem is used by:  reseq12i  5063  resindm  5108  resmpt  5114  resmpt3  5115  resmptf  5116  opabresid  5119  rescnvcnv  5253  coires1  5308  fresaunres1disj  5574  fcoi1  5575  fvsnun1  5915  fvsnun2  5916  resoprab  6187  resmpo  6189  ofmres  6373  f1stres  6397  f2ndres  6398  df1st2  6459  df2nd2  6460  dftpos2  6536  tfr2a  6596  freccllem  6677  frecfcllem  6679  frecsuclem  6681  djuf1olemr  7400  divfnzn  10037  gsummptfidmadd  14251  gsummptfidmsplitres  14255  cnmptid  15507  xmsxmet2  15689  msmet2  15690  cnfldms  15762