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Theorem coss2 5854
Description: Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
Assertion
Ref Expression
coss2 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Proof of Theorem coss2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssbr 5191 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
21anim1d 611 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑦𝐶𝑧) → (𝑥𝐵𝑦𝑦𝐶𝑧)))
32eximdv 1920 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)))
43ssopab2dv 5550 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)})
5 df-co 5684 . 2 (𝐶𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)}
6 df-co 5684 . 2 (𝐶𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)}
74, 5, 63sstr4g 4026 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1781  wss 3947   class class class wbr 5147  {copab 5209  ccom 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-br 5148  df-opab 5210  df-co 5684
This theorem is referenced by:  coeq2  5856  funss  6564  tposss  8208  dftpos4  8226  ttrclco  9709  frmin  9740  frrlem16  9749  rtrclreclem4  15004  tsrdir  18553  mvdco  19307  ustex2sym  23712  ustex3sym  23713  ustneism  23719  trust  23725  utop2nei  23746  neipcfilu  23792  fcoinver  31822  trclubgNEW  42354  trrelsuperrel2dg  42407
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