Theorem ixpssixp 44461

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 411	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
4	1, 3	ralrimi 3250	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5		ss2ixp 8933	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
6	4, 5	syl 17	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 394  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3057   ⊆ wss 3947  Xcixp 8920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-v 3473  df-in 3954  df-ss 3964  df-ixp 8921

This theorem is referenced by:  ioosshoi  46059  iinhoiicclem  46063  iinhoiicc  46064  iunhoiioo  46066  vonioolem2  46071  vonicclem2  46074

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