Theorem iinssf 45040

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

Hypothesis

Ref	Expression
iinssf.1	⊢ Ⅎ𝑥𝐶

Assertion

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

Proof of Theorem iinssf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 5020	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3493	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		ssel 4002	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
4	3	reximi 3090	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
5		iinssf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐶
6	5	nfcri 2900	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐶
7	6	r19.36vf 45038	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
8	4, 7	syl 17	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
9	2, 8	biimtrid 242	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐶))
10	9	ssrdv 4014	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108  Ⅎwnfc 2893  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∩ ciin 5016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-ss 3993  df-iin 5018

This theorem is referenced by:  iinssdf  45041  iinss2d  45062