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Mirrors > Home > NFE Home > Th. List > riinrab | GIF version |
Description: Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
riinrab | ⊢ (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = {y ∈ A ∣ ∀x ∈ X φ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riin0 4039 | . . 3 ⊢ (X = ∅ → (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = A) | |
2 | rzal 3651 | . . . . 5 ⊢ (X = ∅ → ∀x ∈ X φ) | |
3 | 2 | ralrimivw 2698 | . . . 4 ⊢ (X = ∅ → ∀y ∈ A ∀x ∈ X φ) |
4 | rabid2 2788 | . . . 4 ⊢ (A = {y ∈ A ∣ ∀x ∈ X φ} ↔ ∀y ∈ A ∀x ∈ X φ) | |
5 | 3, 4 | sylibr 203 | . . 3 ⊢ (X = ∅ → A = {y ∈ A ∣ ∀x ∈ X φ}) |
6 | 1, 5 | eqtrd 2385 | . 2 ⊢ (X = ∅ → (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = {y ∈ A ∣ ∀x ∈ X φ}) |
7 | ssrab2 3351 | . . . . 5 ⊢ {y ∈ A ∣ φ} ⊆ A | |
8 | 7 | rgenw 2681 | . . . 4 ⊢ ∀x ∈ X {y ∈ A ∣ φ} ⊆ A |
9 | riinn0 4040 | . . . 4 ⊢ ((∀x ∈ X {y ∈ A ∣ φ} ⊆ A ∧ X ≠ ∅) → (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = ∩x ∈ X {y ∈ A ∣ φ}) | |
10 | 8, 9 | mpan 651 | . . 3 ⊢ (X ≠ ∅ → (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = ∩x ∈ X {y ∈ A ∣ φ}) |
11 | iinrab 4028 | . . 3 ⊢ (X ≠ ∅ → ∩x ∈ X {y ∈ A ∣ φ} = {y ∈ A ∣ ∀x ∈ X φ}) | |
12 | 10, 11 | eqtrd 2385 | . 2 ⊢ (X ≠ ∅ → (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = {y ∈ A ∣ ∀x ∈ X φ}) |
13 | 6, 12 | pm2.61ine 2592 | 1 ⊢ (A ∩ ∩x ∈ X {y ∈ A ∣ φ}) = {y ∈ A ∣ ∀x ∈ X φ} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ≠ wne 2516 ∀wral 2614 {crab 2618 ∩ cin 3208 ⊆ wss 3257 ∅c0 3550 ∩ciin 3970 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-iin 3972 |
This theorem is referenced by: (None) |
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