Theorem iinssdf 45972

Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
iinssdf.a	⊢ Ⅎ𝑥𝐴
iinssdf.n	⊢ Ⅎ𝑥𝑋
iinssdf.c	⊢ Ⅎ𝑥𝐶
iinssdf.d	⊢ Ⅎ𝑥𝐷
iinssdf.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
iinssdf.b	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷)
iinssdf.s	⊢ (𝜑 → 𝐷 ⊆ 𝐶)

Assertion

Ref	Expression
iinssdf	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Proof of Theorem iinssdf

Step	Hyp	Ref	Expression
1		iinssdf.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		iinssdf.s	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐶)
3		iinssdf.d	. . . . 5 ⊢ Ⅎ𝑥𝐷
4		iinssdf.c	. . . . 5 ⊢ Ⅎ𝑥𝐶
5	3, 4	nfss 3924	. . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ 𝐶
6		iinssdf.n	. . . 4 ⊢ Ⅎ𝑥𝑋
7		iinssdf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
8		iinssdf.b	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷)
9	8	sseq1d 3962	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ⊆ 𝐶 ↔ 𝐷 ⊆ 𝐶))
10	5, 6, 7, 9	rspcef 45907	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
11	1, 2, 10	syl2anc 596	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
12	4	iinssf 45971	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
13	11, 12	syl 18	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  ∃wrex 3086   ⊆ wss 3899  ∩ ciin 4952

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3916  df-iin 4954

This theorem is used by: (None)