Theorem iinss 2025

Description: Subset implication for an indexed intersection.

Assertion

Ref	Expression
iinss	⊢ (∃x ∈ A B ⊆ C → ∩x ∈ A B ⊆ C)

Distinct variable group(s):   x,C   x,A

Proof of Theorem iinss

Step	Hyp	Ref	Expression
1		19.12 729	. . . 4 ⊢ (∃x∀y(x ∈ A ∧ (y ∈ B → y ∈ C)) → ∀y∃x(x ∈ A ∧ (y ∈ B → y ∈ C)))
2		df-rex 1206	. . . . 5 ⊢ (∃x ∈ A ∀y(y ∈ B → y ∈ C) ↔ ∃x(x ∈ A ∧ ∀y(y ∈ B → y ∈ C)))
3		19.28v 957	. . . . . 6 ⊢ (∀y(x ∈ A ∧ (y ∈ B → y ∈ C)) ↔ (x ∈ A ∧ ∀y(y ∈ B → y ∈ C)))
4	3	biex 733	. . . . 5 ⊢ (∃x∀y(x ∈ A ∧ (y ∈ B → y ∈ C)) ↔ ∃x(x ∈ A ∧ ∀y(y ∈ B → y ∈ C)))
5	2, 4	bitr4 154	. . . 4 ⊢ (∃x ∈ A ∀y(y ∈ B → y ∈ C) ↔ ∃x∀y(x ∈ A ∧ (y ∈ B → y ∈ C)))
6		df-rex 1206	. . . . 5 ⊢ (∃x ∈ A (y ∈ B → y ∈ C) ↔ ∃x(x ∈ A ∧ (y ∈ B → y ∈ C)))
7	6	bial 695	. . . 4 ⊢ (∀y∃x ∈ A (y ∈ B → y ∈ C) ↔ ∀y∃x(x ∈ A ∧ (y ∈ B → y ∈ C)))
8	1, 5, 7	3imtr4 192	. . 3 ⊢ (∃x ∈ A ∀y(y ∈ B → y ∈ C) → ∀y∃x ∈ A (y ∈ B → y ∈ C))
9		r19.36av 1299	. . . . 5 ⊢ (∃x ∈ A (y ∈ B → y ∈ C) → (∀x ∈ A y ∈ B → y ∈ C))
10		visset 1350	. . . . . 6 ⊢ y ∈ V
11		eliin 1999	. . . . . 6 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B))
12	10, 11	ax-mp 6	. . . . 5 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)
13	9, 12	syl5ib 181	. . . 4 ⊢ (∃x ∈ A (y ∈ B → y ∈ C) → (y ∈ ∩x ∈ A B → y ∈ C))
14	13	19.20i 691	. . 3 ⊢ (∀y∃x ∈ A (y ∈ B → y ∈ C) → ∀y(y ∈ ∩x ∈ A B → y ∈ C))
15	8, 14	syl 12	. 2 ⊢ (∃x ∈ A ∀y(y ∈ B → y ∈ C) → ∀y(y ∈ ∩x ∈ A B → y ∈ C))
16		dfss2 1497	. . 3 ⊢ (B ⊆ C ↔ ∀y(y ∈ B → y ∈ C))
17	16	birex 1224	. 2 ⊢ (∃x ∈ A B ⊆ C ↔ ∃x ∈ A ∀y(y ∈ B → y ∈ C))
18		dfss2 1497	. 2 ⊢ (∩x ∈ A B ⊆ C ↔ ∀y(y ∈ ∩x ∈ A B → y ∈ C))
19	15, 17, 18	3imtr4 192	1 ⊢ (∃x ∈ A B ⊆ C → ∩x ∈ A B ⊆ C)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∩ciin 1995

This theorem is referenced by:  scott0 3542

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-iin 1997

metamath.org