Theorem reseq2 4809

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 4548	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3272	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4546	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4546	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2195	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1331  Vcvv 2681   ∩ cin 3065   × cxp 4532   ↾ cres 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-opab 3985  df-xp 4540  df-res 4546

This theorem is referenced by:  reseq2i  4811  reseq2d  4814  resabs1  4843  resima2  4848  imaeq2  4872  resdisj  4962  relcoi1  5065  fressnfv  5600  tfrlem1  6198  tfrlem9  6209  tfr0dm  6212  tfrlemisucaccv  6215  tfrlemiubacc  6220  tfr1onlemsucaccv  6231  tfr1onlemubacc  6236  tfr1onlemaccex  6238  tfrcllemsucaccv  6244  tfrcllembxssdm  6246  tfrcllemubacc  6249  tfrcllemaccex  6251  tfrcllemres  6252  tfrcldm  6253  fnfi  6818  lmbr2  12372  lmff  12407