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Mirrors > Home > ILE Home > Th. List > riota2df | GIF version |
Description: A deduction version of riota2f 5489. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
riota2df.1 | ⊢ Ⅎ𝑥𝜑 |
riota2df.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
riota2df.3 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
riota2df.4 | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
riota2df.5 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
riota2df | ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riota2df.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
2 | 1 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐵 ∈ 𝐴) |
3 | simpr 103 | . . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝜓) | |
4 | df-reu 2313 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | 3, 4 | sylib 127 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
6 | simpr 103 | . . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
7 | 2 | adantr 261 | . . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴) |
8 | 6, 7 | eqeltrd 2114 | . . . . 5 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) |
9 | 8 | biantrurd 289 | . . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
10 | riota2df.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
11 | 10 | adantlr 446 | . . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
12 | 9, 11 | bitr3d 179 | . . 3 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝜒)) |
13 | riota2df.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
14 | nfreu1 2481 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜓 | |
15 | 13, 14 | nfan 1457 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) |
16 | riota2df.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
17 | 16 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝜒) |
18 | riota2df.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
19 | 18 | adantr 261 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝐵) |
20 | 2, 5, 12, 15, 17, 19 | iota2df 4891 | . 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵)) |
21 | df-riota 5468 | . . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
22 | 21 | eqeq1i 2047 | . 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵) |
23 | 20, 22 | syl6bbr 187 | 1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 ∃!weu 1900 Ⅎwnfc 2165 ∃!wreu 2308 ℩cio 4865 ℩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-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-reu 2313 df-v 2559 df-sbc 2765 df-un 2922 df-sn 3381 df-pr 3382 df-uni 3581 df-iota 4867 df-riota 5468 |
This theorem is referenced by: riota2f 5489 riota5f 5492 |
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