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Mirrors > Home > MPE Home > Th. List > Mathboxes > iineq12f | Structured version Visualization version GIF version |
Description: Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
Ref | Expression |
---|---|
iineq12f.1 | ⊢ Ⅎ𝑥𝐴 |
iineq12f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
iineq12f | ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2901 | . . . . . 6 ⊢ (𝐶 = 𝐷 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) | |
2 | 1 | ralimi 3160 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷)) |
3 | ralbi 3167 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 = 𝐷 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷)) |
5 | iineq12f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
6 | iineq12f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | raleqf 3397 | . . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐷 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
8 | 4, 7 | sylan9bbr 513 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷)) |
9 | 8 | abbidv 2885 | . 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷}) |
10 | df-iin 4921 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} | |
11 | df-iin 4921 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐷 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐷} | |
12 | 9, 10, 11 | 3eqtr4g 2881 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐷) → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 ∀wral 3138 ∩ ciin 4919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-iin 4921 |
This theorem is referenced by: (None) |
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