Theorem reseq1i 4954

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 4952	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1372   |` cres 4676

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-res 4686

This theorem is referenced by:  reseq12i  4956  resindm  5000  resmpt  5006  resmpt3  5007  resmptf  5008  opabresid  5011  rescnvcnv  5144  coires1  5199  fcoi1  5455  fvsnun1  5780  fvsnun2  5781  resoprab  6040  resmpo  6042  ofmres  6220  f1stres  6244  f2ndres  6245  df1st2  6304  df2nd2  6305  dftpos2  6346  tfr2a  6406  freccllem  6487  frecfcllem  6489  frecsuclem  6491  djuf1olemr  7155  divfnzn  9741  cnmptid  14724  xmsxmet2  14906  msmet2  14907  cnfldms  14979