Theorem reseq2 4886

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 4625	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3328	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4623	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4623	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
5	2, 3, 4	3eqtr4g 2228	1 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1348  Vcvv 2730   ∩ cin 3120   × cxp 4609   ↾ cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-opab 4051  df-xp 4617  df-res 4623

This theorem is referenced by:  reseq2i  4888  reseq2d  4891  resabs1  4920  resima2  4925  imaeq2  4949  resdisj  5039  relcoi1  5142  fressnfv  5683  tfrlem1  6287  tfrlem9  6298  tfr0dm  6301  tfrlemisucaccv  6304  tfrlemiubacc  6309  tfr1onlemsucaccv  6320  tfr1onlemubacc  6325  tfr1onlemaccex  6327  tfrcllemsucaccv  6333  tfrcllembxssdm  6335  tfrcllemubacc  6338  tfrcllemaccex  6340  tfrcllemres  6341  tfrcldm  6342  fnfi  6914  lmbr2  13008  lmff  13043