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Mirrors > Home > ILE Home > Th. List > riotaprop | GIF version |
Description: Properties of a restricted definite description operator. Todo (df-riota 5738 update): can some uses of riota2f 5759 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
Ref | Expression |
---|---|
riotaprop.0 | ⊢ Ⅎ𝑥𝜓 |
riotaprop.1 | ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑) |
riotaprop.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riotaprop | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaprop.1 | . . 3 ⊢ 𝐵 = (℩𝑥 ∈ 𝐴 𝜑) | |
2 | riotacl 5752 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
3 | 1, 2 | eqeltrid 2227 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴) |
4 | 1 | eqcomi 2144 | . . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵 |
5 | nfriota1 5745 | . . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑) | |
6 | 1, 5 | nfcxfr 2279 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
7 | riotaprop.0 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | riotaprop.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
9 | 6, 7, 8 | riota2f 5759 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
10 | 4, 9 | mpbiri 167 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → 𝜓) |
11 | 3, 10 | mpancom 419 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓) |
12 | 3, 11 | jca 304 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 Ⅎwnf 1437 ∈ wcel 1481 ∃!wreu 2419 ℩crio 5737 |
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-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-uni 3745 df-iota 5096 df-riota 5738 |
This theorem is referenced by: lble 8729 |
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