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Mirrors > Home > ILE Home > Th. List > iinconstm | GIF version |
Description: Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Ref | Expression |
---|---|
iinconstm | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2692 | . . . 4 ⊢ 𝑧 ∈ V | |
2 | eliin 3826 | . . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) |
4 | r19.3rmv 3458 | . . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) | |
5 | 3, 4 | bitr4id 198 | . 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ 𝐵)) |
6 | 5 | eqrdv 2138 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 ∩ ciin 3822 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-iin 3824 |
This theorem is referenced by: iin0imm 4100 |
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