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Theorem coss1 5866
Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
coss1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem coss1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssbr 5187 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑧𝑦𝐵𝑧))
21anim2d 612 . . . 4 (𝐴𝐵 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐶𝑦𝑦𝐵𝑧)))
32eximdv 1917 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
43ssopab2dv 5556 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
5 df-co 5694 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
6 df-co 5694 . 2 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
74, 5, 63sstr4g 4037 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1779  wss 3951   class class class wbr 5143  {copab 5205  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3968  df-br 5144  df-opab 5206  df-co 5694
This theorem is referenced by:  coeq1  5868  funss  6585  tposss  8252  cottrcl  9759  frmin  9789  frrlem16  9798  rtrclreclem4  15100  tsrdir  18649  ustex2sym  24225  ustex3sym  24226  ustneism  24232  trust  24238  utop2nei  24259  neipcfilu  24305  trclubgNEW  43631  trrelsuperrel2dg  43684  trclrelexplem  43724  cotrcltrcl  43738  cotrclrcl  43755  frege96d  43762  frege97d  43765  frege109d  43770  frege131d  43777
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