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Theorem coss1 5869
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 5192 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑧𝑦𝐵𝑧))
21anim2d 612 . . . 4 (𝐴𝐵 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐶𝑦𝑦𝐵𝑧)))
32eximdv 1915 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
43ssopab2dv 5561 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
5 df-co 5698 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
6 df-co 5698 . 2 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
74, 5, 63sstr4g 4041 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wss 3963   class class class wbr 5148  {copab 5210  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ss 3980  df-br 5149  df-opab 5211  df-co 5698
This theorem is referenced by:  coeq1  5871  funss  6587  tposss  8251  cottrcl  9757  frmin  9787  frrlem16  9796  rtrclreclem4  15097  tsrdir  18662  ustex2sym  24241  ustex3sym  24242  ustneism  24248  trust  24254  utop2nei  24275  neipcfilu  24321  trclubgNEW  43608  trrelsuperrel2dg  43661  trclrelexplem  43701  cotrcltrcl  43715  cotrclrcl  43732  frege96d  43739  frege97d  43742  frege109d  43747  frege131d  43754
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