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Mirrors > Home > NFE Home > Th. List > iinrab | GIF version |
Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.) |
Ref | Expression |
---|---|
iinrab | ⊢ (A ≠ ∅ → ∩x ∈ A {y ∈ B ∣ φ} = {y ∈ B ∣ ∀x ∈ A φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.28zv 3645 | . . 3 ⊢ (A ≠ ∅ → (∀x ∈ A (y ∈ B ∧ φ) ↔ (y ∈ B ∧ ∀x ∈ A φ))) | |
2 | 1 | abbidv 2467 | . 2 ⊢ (A ≠ ∅ → {y ∣ ∀x ∈ A (y ∈ B ∧ φ)} = {y ∣ (y ∈ B ∧ ∀x ∈ A φ)}) |
3 | df-rab 2623 | . . . . 5 ⊢ {y ∈ B ∣ φ} = {y ∣ (y ∈ B ∧ φ)} | |
4 | 3 | a1i 10 | . . . 4 ⊢ (x ∈ A → {y ∈ B ∣ φ} = {y ∣ (y ∈ B ∧ φ)}) |
5 | 4 | iineq2i 3988 | . . 3 ⊢ ∩x ∈ A {y ∈ B ∣ φ} = ∩x ∈ A {y ∣ (y ∈ B ∧ φ)} |
6 | iinab 4027 | . . 3 ⊢ ∩x ∈ A {y ∣ (y ∈ B ∧ φ)} = {y ∣ ∀x ∈ A (y ∈ B ∧ φ)} | |
7 | 5, 6 | eqtri 2373 | . 2 ⊢ ∩x ∈ A {y ∈ B ∣ φ} = {y ∣ ∀x ∈ A (y ∈ B ∧ φ)} |
8 | df-rab 2623 | . 2 ⊢ {y ∈ B ∣ ∀x ∈ A φ} = {y ∣ (y ∈ B ∧ ∀x ∈ A φ)} | |
9 | 2, 7, 8 | 3eqtr4g 2410 | 1 ⊢ (A ≠ ∅ → ∩x ∈ A {y ∈ B ∣ φ} = {y ∈ B ∣ ∀x ∈ A φ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {cab 2339 ≠ wne 2516 ∀wral 2614 {crab 2618 ∅c0 3550 ∩ciin 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 df-iin 3972 |
This theorem is referenced by: iinrab2 4029 riinrab 4041 |
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