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Mirrors > Home > ILE Home > Th. List > riotass | GIF version |
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
riotass | ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuss 3218 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) | |
2 | riotasbc 5483 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑) |
4 | simp1 904 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → 𝐴 ⊆ 𝐵) | |
5 | riotacl 5482 | . . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) | |
6 | 1, 5 | syl 14 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
7 | 4, 6 | sseldd 2946 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵) |
8 | simp3 906 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐵 𝜑) | |
9 | nfriota1 5475 | . . . . 5 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 𝜑) | |
10 | 9 | nfsbc1 2781 | . . . . 5 ⊢ Ⅎ𝑥[(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 |
11 | sbceq1a 2773 | . . . . 5 ⊢ (𝑥 = (℩𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑)) | |
12 | 9, 10, 11 | riota2f 5489 | . . . 4 ⊢ (((℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑))) |
13 | 7, 8, 12 | syl2anc 391 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑))) |
14 | 3, 13 | mpbid 135 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = (℩𝑥 ∈ 𝐴 𝜑)) |
15 | 14 | eqcomd 2045 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ∃!wreu 2308 [wsbc 2764 ⊆ wss 2917 ℩crio 5467 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-uni 3581 df-iota 4867 df-riota 5468 |
This theorem is referenced by: moriotass 5496 |
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