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Mirrors > Home > ILE Home > Th. List > riotaprop | GIF version |
Description: Properties of a restricted definite description operator. Todo (df-riota 5622 update): can some uses of riota2f 5643 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 5636 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
3 | 1, 2 | syl5eqel 2175 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴) |
4 | 1 | eqcomi 2093 | . . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵 |
5 | nfriota1 5629 | . . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑) | |
6 | 1, 5 | nfcxfr 2226 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
7 | riotaprop.0 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | riotaprop.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
9 | 6, 7, 8 | riota2f 5643 | . . . 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 414 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓) |
12 | 3, 11 | jca 301 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 Ⅎwnf 1395 ∈ wcel 1439 ∃!wreu 2362 ℩crio 5621 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-uni 3660 df-iota 4993 df-riota 5622 |
This theorem is referenced by: lble 8469 |
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