Theorem coss12d 14907

Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)

Hypotheses

Ref	Expression
coss12d.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
coss12d.c	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Assertion

Ref	Expression
coss12d	⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))

Proof of Theorem coss12d

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coss12d.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
2	1	ssbrd 5143	. . . . 5 ⊢ (𝜑 → (𝑥𝐶𝑦 → 𝑥𝐷𝑦))
3		coss12d.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	ssbrd 5143	. . . . 5 ⊢ (𝜑 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧))
5	2, 4	anim12d 610	. . . 4 ⊢ (𝜑 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)))
6	5	eximdv 1919	. . 3 ⊢ (𝜑 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)))
7	6	ssopab2dv 5507	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)})
8		df-co 5641	. 2 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
9		df-co 5641	. 2 ⊢ (𝐵 ∘ 𝐷) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)}
10	7, 8, 9	3sstr4g 3989	1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781   ⊆ wss 3903   class class class wbr 5100  {copab 5162   ∘ ccom 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-br 5101  df-opab 5163  df-co 5641

This theorem is referenced by:  trrelssd  14908  ustund  24178  bj-imdirco  37439  relexpss1d  44055