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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. TODO (df-riota 6511 update): can some uses of riota2f 6532 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 6525 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
3 | 1, 2 | syl5eqel 2692 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝐵 ∈ 𝐴) |
4 | 1 | eqcomi 2619 | . . . 4 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵 |
5 | nfriota1 6518 | . . . . . 6 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑) | |
6 | 1, 5 | nfcxfr 2749 | . . . . 5 ⊢ Ⅎ𝑥𝐵 |
7 | riotaprop.0 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
8 | riotaprop.2 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
9 | 6, 7, 8 | riota2f 6532 | . . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵)) |
10 | 4, 9 | mpbiri 247 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → 𝜓) |
11 | 3, 10 | mpancom 700 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → 𝜓) |
12 | 3, 11 | jca 553 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∃!wreu 2898 ℩crio 6510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-riota 6511 |
This theorem is referenced by: fin23lem27 9033 lble 10854 ltrniotaval 34887 |
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