MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coss12d Structured version   Visualization version   GIF version

Theorem coss12d 15011
Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)
Hypotheses
Ref Expression
coss12d.a (𝜑𝐴𝐵)
coss12d.c (𝜑𝐶𝐷)
Assertion
Ref Expression
coss12d (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))

Proof of Theorem coss12d
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coss12d.c . . . . . 6 (𝜑𝐶𝐷)
21ssbrd 5186 . . . . 5 (𝜑 → (𝑥𝐶𝑦𝑥𝐷𝑦))
3 coss12d.a . . . . . 6 (𝜑𝐴𝐵)
43ssbrd 5186 . . . . 5 (𝜑 → (𝑦𝐴𝑧𝑦𝐵𝑧))
52, 4anim12d 609 . . . 4 (𝜑 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐷𝑦𝑦𝐵𝑧)))
65eximdv 1917 . . 3 (𝜑 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)))
76ssopab2dv 5556 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)})
8 df-co 5694 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
9 df-co 5694 . 2 (𝐵𝐷) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)}
107, 8, 93sstr4g 4037 1 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1779  wss 3951   class class class wbr 5143  {copab 5205  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3968  df-br 5144  df-opab 5206  df-co 5694
This theorem is referenced by:  trrelssd  15012  ustund  24230  bj-imdirco  37191  relexpss1d  43718
  Copyright terms: Public domain W3C validator