Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iinssdf | Structured version Visualization version GIF version |
Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
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) |
Ref | Expression |
---|---|
iinssdf | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
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 3596 | . . . 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 3632 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ⊆ 𝐶 ↔ 𝐷 ⊆ 𝐶)) |
10 | 5, 6, 7, 9 | rspcef 39241 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
11 | 1, 2, 10 | syl2anc 693 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
12 | 4 | iinssf 39327 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
13 | 11, 12 | syl 17 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Ⅎwnfc 2751 ∃wrex 2913 ⊆ 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-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-iin 4523 |
This theorem is referenced by: (None) |
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