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Mirrors > Home > MPE Home > Th. List > Mathboxes > intabssd | Structured version Visualization version GIF version |
Description: When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.) |
Ref | Expression |
---|---|
intabssd.ex | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
intabssd.sub | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
intabssd.ss | ⊢ (𝜑 → 𝐴 ⊆ 𝑦) |
Ref | Expression |
---|---|
intabssd | ⊢ (𝜑 → ∩ {𝑥 ∣ 𝜓} ⊆ ∩ {𝑦 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intabssd.ex | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | intabssd.sub | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) | |
3 | eleq2 2829 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴)) | |
4 | 3 | biimpd 228 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝐴)) |
5 | intabssd.ss | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑦) | |
6 | 5 | sseld 3925 | . . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑦)) |
7 | 4, 6 | sylan9r 509 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
8 | 2, 7 | imim12d 81 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝜓 → 𝑧 ∈ 𝑥) → (𝜒 → 𝑧 ∈ 𝑦))) |
9 | 1, 8 | spcimdv 3531 | . . . 4 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑧 ∈ 𝑥) → (𝜒 → 𝑧 ∈ 𝑦))) |
10 | 9 | alrimdv 1936 | . . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑧 ∈ 𝑥) → ∀𝑦(𝜒 → 𝑧 ∈ 𝑦))) |
11 | vex 3435 | . . . 4 ⊢ 𝑧 ∈ V | |
12 | 11 | elintab 4896 | . . 3 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜓 → 𝑧 ∈ 𝑥)) |
13 | 11 | elintab 4896 | . . 3 ⊢ (𝑧 ∈ ∩ {𝑦 ∣ 𝜒} ↔ ∀𝑦(𝜒 → 𝑧 ∈ 𝑦)) |
14 | 10, 12, 13 | 3imtr4g 296 | . 2 ⊢ (𝜑 → (𝑧 ∈ ∩ {𝑥 ∣ 𝜓} → 𝑧 ∈ ∩ {𝑦 ∣ 𝜒})) |
15 | 14 | ssrdv 3932 | 1 ⊢ (𝜑 → ∩ {𝑥 ∣ 𝜓} ⊆ ∩ {𝑦 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1540 = wceq 1542 ∈ wcel 2110 {cab 2717 ⊆ wss 3892 ∩ cint 4885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-v 3433 df-in 3899 df-ss 3909 df-int 4886 |
This theorem is referenced by: harval3 41134 clcnvlem 41213 |
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