Theorem reseq2 4999

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 4732	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3405	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4730	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4730	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1395  Vcvv 2799   ∩ cin 3196   × cxp 4716   ↾ cres 4720

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-opab 4145  df-xp 4724  df-res 4730

This theorem is referenced by:  reseq2i  5001  reseq2d  5004  resabs1  5033  resima2  5038  imaeq2  5063  resdisj  5156  relcoi1  5259  fressnfv  5825  tfrlem1  6452  tfrlem9  6463  tfr0dm  6466  tfrlemisucaccv  6469  tfrlemiubacc  6474  tfr1onlemsucaccv  6485  tfr1onlemubacc  6490  tfr1onlemaccex  6492  tfrcllemsucaccv  6498  tfrcllembxssdm  6500  tfrcllemubacc  6503  tfrcllemaccex  6505  tfrcllemres  6506  tfrcldm  6507  fnfi  7099  lmbr2  14882  lmff  14917  dvmptid  15384