Theorem reseq2 5391

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 5128	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3814	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 5126	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 5126	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2681	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1483  Vcvv 3200   ∩ cin 3573   × cxp 5112   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-opab 4713  df-xp 5120  df-res 5126

This theorem is referenced by:  reseq2i  5393  reseq2d  5396  resabs1  5427  resima2  5432  resima2OLD  5433  imaeq2  5462  resdisj  5563  fressnfv  6427  tfrlem1  7472  tfrlem9  7481  tfrlem11  7484  tfrlem12  7485  tfr2b  7492  tz7.44-1  7502  tz7.44-2  7503  tz7.44-3  7504  rdglem1  7511  fnfi  8238  fseqenlem1  8847  rtrclreclem4  13801  psgnprfval1  17942  gsumzaddlem  18321  gsum2dlem2  18370  znunithash  19913  islinds  20148  lmbr2  21063  lmff  21105  kgencn2  21360  ptcmpfi  21616  tsmsgsum  21942  tsmsres  21947  tsmsf1o  21948  tsmsxplem1  21956  tsmsxp  21958  ustval  22006  xrge0gsumle  22636  xrge0tsms  22637  lmmbr2  23057  lmcau  23111  limcun  23659  jensen  24715  wilthlem2  24795  wilthlem3  24796  hhssnvt  28122  hhsssh  28126  foresf1o  29343  gsumle  29779  xrge0tsmsd  29785  esumsnf  30126  subfacp1lem3  31164  subfacp1lem5  31166  erdszelem1  31173  erdsze  31184  erdsze2lem2  31186  cvmscbv  31240  cvmshmeo  31253  cvmsss2  31256  dfpo2  31645  eldm3  31651  dfrdg2  31701  mbfresfi  33456  mzpcompact2lem  37314  seff  38508  wessf1ornlem  39371  fouriersw  40448  sge0tsms  40597  sge0f1o  40599  sge0sup  40608  meadjuni  40674  ismeannd  40684  psmeasurelem  40687  psmeasure  40688  omeunile  40719  isomennd  40745  hoidmvlelem3  40811