Theorem ixpssixp 39790

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 397	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ⊆ 𝐶))
4	1, 3	ralrimi 3106	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5		ss2ixp 8075	. 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 382  Ⅎwnf 1856   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723  Xcixp 8062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-in 3730  df-ss 3737  df-ixp 8063

This theorem is referenced by:  ioosshoi  41403  iinhoiicclem  41407  iinhoiicc  41408  iunhoiioo  41410  vonioolem2  41415  vonicclem2  41418