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Theorem coss12d 14312
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 5085 . . . . 5 (𝜑 → (𝑥𝐶𝑦𝑥𝐷𝑦))
3 coss12d.a . . . . . 6 (𝜑𝐴𝐵)
43ssbrd 5085 . . . . 5 (𝜑 → (𝑦𝐴𝑧𝑦𝐵𝑧))
52, 4anim12d 610 . . . 4 (𝜑 → ((𝑥𝐶𝑦𝑦𝐴𝑧) → (𝑥𝐷𝑦𝑦𝐵𝑧)))
65eximdv 1918 . . 3 (𝜑 → (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)))
76ssopab2dv 5414 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)})
8 df-co 5540 . 2 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
9 df-co 5540 . 2 (𝐵𝐷) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦𝑦𝐵𝑧)}
107, 8, 93sstr4g 3991 1 (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wex 1780  wss 3913   class class class wbr 5042  {copab 5104  ccom 5535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-v 3475  df-in 3920  df-ss 3930  df-br 5043  df-opab 5105  df-co 5540
This theorem is referenced by:  trrelssd  14313  ustund  22806  relexpss1d  40185
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