Theorem resexg 2597

Description: The restriction of a set is a set.

Assertion

Ref	Expression
resexg	⊢ (A ∈ C → (A ↾ B) ∈ V)

Proof of Theorem resexg

Step	Hyp	Ref	Expression
1		inex1g 1699	. 2 ⊢ (A ∈ C → (A ∩ (B × V)) ∈ V)
2		df-res 2430	. . 3 ⊢ (A ↾ B) = (A ∩ (B × V))
3	2	eleq1i 1152	. 2 ⊢ ((A ↾ B) ∈ V ↔ (A ∩ (B × V)) ∈ V)
4	1, 3	sylibr 175	1 ⊢ (A ∈ C → (A ↾ B) ∈ V)

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  mapunen 3397  php3 3411  ssfi 3430  ruclem5 4889

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-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

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-res 2430

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