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Theorem ssres2 4991
 Description: Subclass theorem for restriction. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 22-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
ssres2 (A B → (C A) (C B))

Proof of Theorem ssres2
StepHypRef Expression
1 xpss1 4856 . . 3 (A B → (A × V) (B × V))
2 sslin 3481 . . 3 ((A × V) (B × V) → (C ∩ (A × V)) (C ∩ (B × V)))
31, 2syl 15 . 2 (A B → (C ∩ (A × V)) (C ∩ (B × V)))
4 df-res 4788 . 2 (C A) = (C ∩ (A × V))
5 df-res 4788 . 2 (C B) = (C ∩ (B × V))
63, 4, 53sstr4g 3312 1 (A B → (C A) (C B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257   × cxp 4770   ↾ cres 4774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-xp 4784  df-res 4788 This theorem is referenced by:  imass2  5024
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