Theorem iinssdf 41776

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 3907	. . . 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 3946	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ⊆ 𝐶 ↔ 𝐷 ⊆ 𝐶))
10	5, 6, 7, 9	rspcef 41706	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
11	1, 2, 10	syl2anc 587	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
12	4	iinssf 41775	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
13	11, 12	syl 17	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∃wrex 3107   ⊆ wss 3881  ∩ ciin 4882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-iin 4884

This theorem is referenced by: (None)