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Theorem coss1 5416
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 4830 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑧𝑦𝐵𝑧))
21anim2d 599 . . . 4 (𝐴𝐵 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐶𝑦𝑦𝐵𝑧)))
32eximdv 1998 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
43ssopab2dv 5137 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
5 df-co 5258 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
6 df-co 5258 . 2 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
74, 5, 63sstr4g 3795 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1852  wss 3723   class class class wbr 4786  {copab 4846  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-in 3730  df-ss 3737  df-br 4787  df-opab 4847  df-co 5258
This theorem is referenced by:  coeq1  5418  funss  6050  tposss  7505  rtrclreclem4  14009  tsrdir  17446  ustex2sym  22240  ustex3sym  22241  ustund  22245  ustneism  22247  trust  22253  utop2nei  22274  neipcfilu  22320  trclubgNEW  38451  trrelsuperrel2dg  38489  trclrelexplem  38529  cotrcltrcl  38543  cotrclrcl  38560  frege96d  38567  frege97d  38570  frege109d  38575  frege131d  38582
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