Theorem coss12dOLD 15168

Description: Obsolete version of coss12d 5845 as of 8-Nov-2026. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
coss12dOLD.a	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
coss12dOLD.c	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Assertion

Ref	Expression
coss12dOLD	⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))

Proof of Theorem coss12dOLD

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

Step	Hyp	Ref	Expression
1		coss12dOLD.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)
2	1	ssbrd 5154	. . . . 5 ⊢ (𝜑 → (𝑥𝐶𝑦 → 𝑥𝐷𝑦))
3		coss12dOLD.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	ssbrd 5154	. . . . 5 ⊢ (𝜑 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧))
5	2, 4	anim12d 621	. . . 4 ⊢ (𝜑 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)))
6	5	eximdv 1952	. . 3 ⊢ (𝜑 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)))
7	6	ssopab2dv 5533	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)})
8		df-co 5667	. 2 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
9		df-co 5667	. 2 ⊢ (𝐵 ∘ 𝐷) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐷𝑦 ∧ 𝑦𝐵𝑧)}
10	7, 8, 9	3sstr4g 3966	1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ⊆ wss 3900   class class class wbr 5108  {copab 5173   ∘ ccom 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-br 5109  df-opab 5174  df-co 5667

This theorem is used by: (None)