Theorem reseq2 4884

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 4623	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
2	1	ineq2d 3328	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3		df-res 4621	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
4		df-res 4621	. 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 4607   ↾ cres 4611

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 4049  df-xp 4615  df-res 4621

This theorem is referenced by:  reseq2i  4886  reseq2d  4889  resabs1  4918  resima2  4923  imaeq2  4947  resdisj  5037  relcoi1  5140  fressnfv  5681  tfrlem1  6285  tfrlem9  6296  tfr0dm  6299  tfrlemisucaccv  6302  tfrlemiubacc  6307  tfr1onlemsucaccv  6318  tfr1onlemubacc  6323  tfr1onlemaccex  6325  tfrcllemsucaccv  6331  tfrcllembxssdm  6333  tfrcllemubacc  6336  tfrcllemaccex  6338  tfrcllemres  6339  tfrcldm  6340  fnfi  6911  lmbr2  12969  lmff  13004