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Mirrors > Home > ILE Home > Th. List > uniiun | GIF version |
Description: Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.) |
Ref | Expression |
---|---|
uniiun | ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfuni2 3582 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
2 | df-iun 3659 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
3 | 1, 2 | eqtr4i 2063 | 1 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 {cab 2026 ∃wrex 2307 ∪ cuni 3580 ∪ 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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 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-rex 2312 df-uni 3581 df-iun 3659 |
This theorem is referenced by: iunpwss 3743 truni 3868 iunpw 4211 reluni 4460 rnuni 4735 imauni 5400 |
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