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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 | r19.3rmv 3423 | . . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) | |
2 | vex 2663 | . . . 4 ⊢ 𝑧 ∈ V | |
3 | eliin 3788 | . . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) |
5 | 1, 4 | syl6rbbr 198 | . 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ 𝐵)) |
6 | 5 | eqrdv 2115 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∃wex 1453 ∈ wcel 1465 ∀wral 2393 Vcvv 2660 ∩ ciin 3784 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-iin 3786 |
This theorem is referenced by: iin0imm 4062 |
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