Theorem reseq2 5851

Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Assertion

Ref	Expression
reseq2	⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Proof of Theorem reseq2

Step	Hyp	Ref	Expression
1		xpeq1 5572	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 4192	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 5570	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 5570	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2884	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1536  Vcvv 3497   ∩ cin 3938   × cxp 5556   ↾ cres 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3150  df-in 3946  df-opab 5132  df-xp 5564  df-res 5570

This theorem is referenced by:  reseq2i  5853  reseq2d  5856  resabs1  5886  resima2  5891  imaeq2  5928  resdisj  6029  fimadmfoALT  6604  fressnfv  6925  tfrlem1  8015  tfrlem9  8024  tfrlem11  8027  tfrlem12  8028  tfr2b  8035  tz7.44-1  8045  tz7.44-2  8046  tz7.44-3  8047  rdglem1  8054  fnfi  8799  fseqenlem1  9453  rtrclreclem4  14423  psgnprfval1  18653  gsumzaddlem  19044  gsum2dlem2  19094  znunithash  20714  islinds  20956  lmbr2  21870  lmff  21912  kgencn2  22168  ptcmpfi  22424  tsmsgsum  22750  tsmsres  22755  tsmsf1o  22756  tsmsxplem1  22764  tsmsxp  22766  ustval  22814  xrge0gsumle  23444  xrge0tsms  23445  lmmbr2  23865  lmcau  23919  limcun  24496  jensen  25569  wilthlem2  25649  wilthlem3  25650  hhssnvt  29045  hhsssh  29049  foresf1o  30268  reldisjun  30356  xrge0tsmsd  30696  gsumle  30729  esumsnf  31327  subfacp1lem3  32433  subfacp1lem5  32435  erdszelem1  32442  erdsze  32453  erdsze2lem2  32455  cvmscbv  32509  cvmshmeo  32522  cvmsss2  32525  dfpo2  32995  eldm3  33001  dfrdg2  33044  bj-diagval  34470  mbfresfi  34942  elcoeleqvrels  35834  eleldisjs  35965  eldisjeq  35978  mzpcompact2lem  39354  seff  40647  wessf1ornlem  41451  fouriersw  42523  sge0tsms  42669  sge0f1o  42671  sge0sup  42680  meadjuni  42746  ismeannd  42756  psmeasurelem  42759  psmeasure  42760  omeunile  42794  isomennd  42820  hoidmvlelem3  42886