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Theorem coss1 5493
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 4899 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑧𝑦𝐵𝑧))
21anim2d 601 . . . 4 (𝐴𝐵 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐶𝑦𝑦𝐵𝑧)))
32eximdv 2008 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
43ssopab2dv 5212 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
5 df-co 5333 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
6 df-co 5333 . 2 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
74, 5, 63sstr4g 3854 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1859  wss 3780   class class class wbr 4855  {copab 4917  ccom 5328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-in 3787  df-ss 3794  df-br 4856  df-opab 4918  df-co 5333
This theorem is referenced by:  coeq1  5495  funss  6130  tposss  7598  rtrclreclem4  14044  tsrdir  17463  ustex2sym  22254  ustex3sym  22255  ustund  22259  ustneism  22261  trust  22267  utop2nei  22288  neipcfilu  22334  trclubgNEW  38443  trrelsuperrel2dg  38481  trclrelexplem  38521  cotrcltrcl  38535  cotrclrcl  38552  frege96d  38559  frege97d  38562  frege109d  38567  frege131d  38574
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