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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iinssd | Structured version Visualization version GIF version |
Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
iinssd.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
iinssd.2 | ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷) |
iinssd.3 | ⊢ (𝜑 → 𝐷 ⊆ 𝐶) |
Ref | Expression |
---|---|
iinssd | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iinssd.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
2 | iinssd.3 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐶) | |
3 | iinssd.2 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷) | |
4 | 3 | sseq1d 4008 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ⊆ 𝐶 ↔ 𝐷 ⊆ 𝐶)) |
5 | 4 | rspcev 3606 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
6 | 1, 2, 5 | syl2anc 583 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
7 | iinss 5052 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ⊆ wss 3943 ∩ ciin 4991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-v 3470 df-in 3950 df-ss 3960 df-iin 4993 |
This theorem is referenced by: smfsuplem3 46082 smflimsuplem1 46089 |
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