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Theorem coss2 4580
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 3878 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
32anim1d 329 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑦𝐶𝑧) → (𝑥𝐵𝑦𝑦𝐶𝑧)))
43eximdv 1808 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)))
54ssopab2dv 4096 . 2 (𝐴𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)})
6 df-co 4437 . 2 (𝐶𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦𝑦𝐶𝑧)}
7 df-co 4437 . 2 (𝐶𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦𝑦𝐶𝑧)}
85, 6, 73sstr4g 3065 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1426  wss 2997   class class class wbr 3837  {copab 3890  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3003  df-ss 3010  df-br 3838  df-opab 3892  df-co 4437
This theorem is referenced by:  coeq2  4582  funss  5020  tposss  5993  dftpos4  6010
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