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

Theorem coss12d 13692
 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 4687 . . . . 5 (𝜑 → (𝑥𝐶𝑦𝑥𝐷𝑦))
3 coss12d.a . . . . . 6 (𝜑𝐴𝐵)
43ssbrd 4687 . . . . 5 (𝜑 → (𝑦𝐴𝑧𝑦𝐵𝑧))
52, 4anim12d 585 . . . 4 (𝜑 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐷𝑦𝑦𝐵𝑧)))
65eximdv 1844 . . 3 (𝜑 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)))
76ssopab2dv 4994 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)})
8 df-co 5113 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
9 df-co 5113 . 2 (𝐵𝐷) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)}
107, 8, 93sstr4g 3638 1 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1702   ⊆ wss 3567   class class class wbr 4644  {copab 4703   ∘ ccom 5108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-in 3574  df-ss 3581  df-br 4645  df-opab 4704  df-co 5113 This theorem is referenced by:  trrelssd  13693  relexpss1d  37816
 Copyright terms: Public domain W3C validator