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| Mirrors > Home > MPE Home > Th. List > riotaprop | Structured version Visualization version GIF version | ||
| Description: Properties of a restricted definite description operator. (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 7405 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
| 3 | 1, 2 | eqeltrid 2845 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴) |
| 4 | 1 | eqcomi 2746 | . . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵 |
| 5 | nfriota1 7395 | . . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑) | |
| 6 | 1, 5 | nfcxfr 2903 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
| 7 | riotaprop.0 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 8 | riotaprop.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 9 | 6, 7, 8 | riota2f 7412 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
| 10 | 4, 9 | mpbiri 258 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → 𝜓) |
| 11 | 3, 10 | mpancom 688 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓) |
| 12 | 3, 11 | jca 511 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∃!wreu 3378 ℩crio 7387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-riota 7388 |
| This theorem is referenced by: fin23lem27 10368 lble 12220 ltrniotaval 40583 |
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