Theorem ixpssixp 46246

Description: Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
ixpssixp.1	⊢ Ⅎ𝑥𝜑
ixpssixp.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
ixpssixp	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)

Proof of Theorem ixpssixp

Step	Hyp	Ref	Expression
1		ixpssixp.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ixpssixp.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
3	2	ex 418	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
4	1, 3	ralrimi 3263	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5		ss2ixp 8949	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
6	4, 5	syl 18	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1815   ∈ wcel 2146  ∀wral 3079   ⊆ wss 3899  Xcixp 8936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-ss 3900  df-ixp 8937

This theorem is used by:  ioosshoi  47818  iinhoiicclem  47822  iinhoiicc  47823  iunhoiioo  47825  vonioolem2  47830  vonicclem2  47833