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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 7379 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
3 | 1, 2 | eqeltrid 2831 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴) |
4 | 1 | eqcomi 2735 | . . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵 |
5 | nfriota1 7368 | . . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑) | |
6 | 1, 5 | nfcxfr 2895 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
7 | riotaprop.0 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | riotaprop.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
9 | 6, 7, 8 | riota2f 7386 | . . . 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 685 | . 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 205 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∃!wreu 3368 ℩crio 7360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-un 3948 df-in 3950 df-ss 3960 df-sn 4624 df-pr 4626 df-uni 4903 df-iota 6489 df-riota 7361 |
This theorem is referenced by: fin23lem27 10325 lble 12170 ltrniotaval 39965 |
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