Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iinssiin | Structured version Visualization version GIF version |
Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
iinssiin.1 | ⊢ Ⅎ𝑥𝜑 |
iinssiin.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
iinssiin | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinssiin.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
2 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
3 | nfii1 4551 | . . . . . . 7 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | |
4 | 2, 3 | nfel 2777 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 |
5 | 1, 4 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) |
6 | iinssiin.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) | |
7 | 6 | adantlr 751 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) |
8 | eliinid 39294 | . . . . . . . 8 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) | |
9 | 8 | adantll 750 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵) |
10 | 7, 9 | sseldd 3604 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐶) |
11 | 10 | ex 450 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐶)) |
12 | 5, 11 | ralrimi 2957 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) |
13 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
14 | eliin 4525 | . . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) |
16 | 12, 15 | sylibr 224 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) |
17 | 16 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) |
18 | dfss3 3592 | . 2 ⊢ (∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) | |
19 | 17, 18 | sylibr 224 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ∩ ciin 4521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-in 3581 df-ss 3588 df-iin 4523 |
This theorem is referenced by: (None) |
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