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Mirrors > Home > ILE 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 3919 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} | |
2 | df-rab 2457 | . . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)}) |
4 | 3 | iuneq2i 3891 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜑)} |
5 | df-rab 2457 | . . 3 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)} | |
6 | r19.42v 2627 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)) | |
7 | 6 | abbii 2286 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)} |
8 | 5, 7 | eqtr4i 2194 | . 2 ⊢ {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)} |
9 | 1, 4, 8 | 3eqtr4i 2201 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∈ wcel 2141 {cab 2156 ∃wrex 2449 {crab 2452 ∪ ciun 3873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-in 3127 df-ss 3134 df-iun 3875 |
This theorem is referenced by: hashrabrex 11444 phisum 12194 |
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