Theorem reseq2 4930

Description: Equality theorem for restrictions. (Contributed by set.mm contributors, 8-Aug-1994.)

Assertion

Ref	Expression
reseq2	⊢ (A = B → (C ↾ A) = (C ↾ B))

Proof of Theorem reseq2

Step	Hyp	Ref	Expression
1		xpeq1 4799	. . 3 ⊢ (A = B → (A × V) = (B × V))
2	1	ineq2d 3458	. 2 ⊢ (A = B → (C ∩ (A × V)) = (C ∩ (B × V)))
3		df-res 4789	. 2 ⊢ (C ↾ A) = (C ∩ (A × V))
4		df-res 4789	. 2 ⊢ (C ↾ B) = (C ∩ (B × V))
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → (C ↾ A) = (C ↾ B))

Syntax hints:   → wi 4   = wceq 1642  Vcvv 2860   ∩ cin 3209   × cxp 4771   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-opab 4624  df-xp 4785  df-res 4789

This theorem is referenced by:  reseq2i  4932  reseq2d  4935  resdisj  5051  fressnfv  5440