Theorem reseq2 2576

Description: Equality theorem for restrictions.

Assertion

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

Proof of Theorem reseq2

Step	Hyp	Ref	Expression
1		xpeq1 2440	. . 3 ⊢ (A = B → (A × V) = (B × V))
2	1	ineq2d 1645	. 2 ⊢ (A = B → (C ∩ (A × V)) = (C ∩ (B × V)))
3		df-res 2430	. 2 ⊢ (C ↾ A) = (C ∩ (A × V))
4		df-res 2430	. 2 ⊢ (C ↾ B) = (C ∩ (B × V))
5	2, 3, 4	3eqtr4g 1147	1 ⊢ (A = B → (C ↾ A) = (C ↾ B))

Syntax hints:   → wi 2   = wceq 1091  Vcvv 1348   ∩ cin 1486   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  imaeq2 2603  funcnvres 2710  f1ococnv2 2817  fnressn 2897  fressnfv 2898  tfrlem11 2959  tfr2 2963  tz7.44-1 2966  tz7.44-2 2967  tz7.44-3 2968  rdglem1 2975  sbthlem4 3352

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-opab 2098  df-xp 2424  df-res 2430

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