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Mirrors > Home > ILE Home > Th. List > iunconstm | GIF version |
Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Jim Kingdon, 15-Aug-2018.) |
Ref | Expression |
---|---|
iunconstm | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.9rmv 3313 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
2 | eliun 3661 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
3 | 1, 2 | syl6rbbr 188 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵)) |
4 | 3 | eqrdv 2038 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 ∪ ciun 3657 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-iun 3659 |
This theorem is referenced by: (None) |
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