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Mirrors > Home > NFE Home > Th. List > iunrab | GIF version |
Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
iunrab | ⊢ ∪x ∈ A {y ∈ B ∣ φ} = {y ∈ B ∣ ∃x ∈ A φ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunab 4012 | . 2 ⊢ ∪x ∈ A {y ∣ (y ∈ B ∧ φ)} = {y ∣ ∃x ∈ A (y ∈ B ∧ φ)} | |
2 | df-rab 2623 | . . . 4 ⊢ {y ∈ B ∣ φ} = {y ∣ (y ∈ B ∧ φ)} | |
3 | 2 | a1i 10 | . . 3 ⊢ (x ∈ A → {y ∈ B ∣ φ} = {y ∣ (y ∈ B ∧ φ)}) |
4 | 3 | iuneq2i 3987 | . 2 ⊢ ∪x ∈ A {y ∈ B ∣ φ} = ∪x ∈ A {y ∣ (y ∈ B ∧ φ)} |
5 | df-rab 2623 | . . 3 ⊢ {y ∈ B ∣ ∃x ∈ A φ} = {y ∣ (y ∈ B ∧ ∃x ∈ A φ)} | |
6 | r19.42v 2765 | . . . 4 ⊢ (∃x ∈ A (y ∈ B ∧ φ) ↔ (y ∈ B ∧ ∃x ∈ A φ)) | |
7 | 6 | abbii 2465 | . . 3 ⊢ {y ∣ ∃x ∈ A (y ∈ B ∧ φ)} = {y ∣ (y ∈ B ∧ ∃x ∈ A φ)} |
8 | 5, 7 | eqtr4i 2376 | . 2 ⊢ {y ∈ B ∣ ∃x ∈ A φ} = {y ∣ ∃x ∈ A (y ∈ B ∧ φ)} |
9 | 1, 4, 8 | 3eqtr4i 2383 | 1 ⊢ ∪x ∈ A {y ∈ B ∣ φ} = {y ∈ B ∣ ∃x ∈ A φ} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2615 {crab 2618 ∪ciun 3969 |
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-ral 2619 df-rex 2620 df-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-iun 3971 |
This theorem is referenced by: (None) |
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