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Theorem coss2 4633
 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 id 19 . . . . . 6 (𝐴𝐵𝐴𝐵)
21ssbrd 3916 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
32anim1d 332 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑦𝐶𝑧) → (𝑥𝐵𝑦𝑦𝐶𝑧)))
43eximdv 1819 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)))
54ssopab2dv 4138 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)})
6 df-co 4486 . 2 (𝐶𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)}
7 df-co 4486 . 2 (𝐶𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)}
85, 6, 73sstr4g 3090 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1436   ⊆ wss 3021   class class class wbr 3875  {copab 3928   ∘ ccom 4481 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-in 3027  df-ss 3034  df-br 3876  df-opab 3930  df-co 4486 This theorem is referenced by:  coeq2  4635  funss  5078  tposss  6073  dftpos4  6090
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