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Theorem coss12d 15008
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 5191 . . . . 5 (𝜑 → (𝑥𝐶𝑦𝑥𝐷𝑦))
3 coss12d.a . . . . . 6 (𝜑𝐴𝐵)
43ssbrd 5191 . . . . 5 (𝜑 → (𝑦𝐴𝑧𝑦𝐵𝑧))
52, 4anim12d 609 . . . 4 (𝜑 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐷𝑦𝑦𝐵𝑧)))
65eximdv 1915 . . 3 (𝜑 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)))
76ssopab2dv 5561 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)})
8 df-co 5698 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
9 df-co 5698 . 2 (𝐵𝐷) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)}
107, 8, 93sstr4g 4041 1 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wss 3963   class class class wbr 5148  {copab 5210  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ss 3980  df-br 5149  df-opab 5211  df-co 5698
This theorem is referenced by:  trrelssd  15009  ustund  24246  bj-imdirco  37173  relexpss1d  43695
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