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Theorem coss2 5311
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 4729 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
21anim1d 587 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑦𝐶𝑧) → (𝑥𝐵𝑦𝑦𝐶𝑧)))
32eximdv 1886 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)))
43ssopab2dv 5033 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)})
5 df-co 5152 . 2 (𝐶𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)}
6 df-co 5152 . 2 (𝐶𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)}
74, 5, 63sstr4g 3679 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1744  wss 3607   class class class wbr 4685  {copab 4745  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-in 3614  df-ss 3621  df-br 4686  df-opab 4746  df-co 5152
This theorem is referenced by:  coeq2  5313  funss  5945  tposss  7398  dftpos4  7416  rtrclreclem4  13845  tsrdir  17285  mvdco  17911  ustex2sym  22067  ustex3sym  22068  ustund  22072  ustneism  22074  trust  22080  utop2nei  22101  neipcfilu  22147  fcoinver  29544  trclubgNEW  38242  trrelsuperrel2dg  38280
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