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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 3306 | . . 3 ⊢ (∃y y ∈ A → (z ∈ B ↔ ∀x ∈ A z ∈ B)) | |
2 | vex 2554 | . . . 4 ⊢ z ∈ V | |
3 | eliin 3653 | . . . 4 ⊢ (z ∈ V → (z ∈ ∩ x ∈ A B ↔ ∀x ∈ A z ∈ B)) | |
4 | 2, 3 | ax-mp 7 | . . 3 ⊢ (z ∈ ∩ x ∈ A B ↔ ∀x ∈ A z ∈ B) |
5 | 1, 4 | syl6rbbr 188 | . 2 ⊢ (∃y y ∈ A → (z ∈ ∩ x ∈ A B ↔ z ∈ B)) |
6 | 5 | eqrdv 2035 | 1 ⊢ (∃y y ∈ A → ∩ x ∈ A B = B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 Vcvv 2551 ∩ ciin 3649 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-iin 3651 |
This theorem is referenced by: iin0imm 3912 |
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