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Theorem coss1 4872
 Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
coss1 (A B → (A C) (B C))

Proof of Theorem coss1
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6 (A BA B)
21ssbrd 4680 . . . . 5 (A B → (yAzyBz))
32anim2d 548 . . . 4 (A B → ((xCy yAz) → (xCy yBz)))
43eximdv 1622 . . 3 (A B → (y(xCy yAz) → y(xCy yBz)))
54ssopab2dv 4715 . 2 (A B → {x, z y(xCy yAz)} {x, z y(xCy yBz)})
6 df-co 4726 . 2 (A C) = {x, z y(xCy yAz)}
7 df-co 4726 . 2 (B C) = {x, z y(xCy yBz)}
85, 6, 73sstr4g 3312 1 (A B → (A C) (B C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   ⊆ wss 3257  {copab 4622   class class class wbr 4639   ∘ ccom 4721 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-br 4640  df-co 4726 This theorem is referenced by:  coeq1  4874  funss  5126
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