Theorem reseq2 4770

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 4511	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3241	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4509	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4509	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2170	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1312  Vcvv 2655   ∩ cin 3034   × cxp 4495   ↾ cres 4499

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-opab 3948  df-xp 4503  df-res 4509

This theorem is referenced by:  reseq2i  4772  reseq2d  4775  resabs1  4804  resima2  4809  imaeq2  4833  resdisj  4923  relcoi1  5026  fressnfv  5559  tfrlem1  6157  tfrlem9  6168  tfr0dm  6171  tfrlemisucaccv  6174  tfrlemiubacc  6179  tfr1onlemsucaccv  6190  tfr1onlemubacc  6195  tfr1onlemaccex  6197  tfrcllemsucaccv  6203  tfrcllembxssdm  6205  tfrcllemubacc  6208  tfrcllemaccex  6210  tfrcllemres  6211  tfrcldm  6212  fnfi  6775  lmbr2  12219  lmff  12254