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Mirrors > Home > MPE Home > Th. List > riota2df | Structured version Visualization version GIF version |
Description: A deduction version of riota2f 7407. (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 479 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐵 ∈ 𝐴) |
3 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝜓) | |
4 | df-reu 3375 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | 3, 4 | sylib 217 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
6 | simpr 483 | . . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
7 | 2 | adantr 479 | . . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴) |
8 | 6, 7 | eqeltrd 2829 | . . . . 5 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) |
9 | 8 | biantrurd 531 | . . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
10 | riota2df.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
11 | 10 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
12 | 9, 11 | bitr3d 280 | . . 3 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝜒)) |
13 | riota2df.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
14 | nfreu1 3406 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜓 | |
15 | 13, 14 | nfan 1894 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) |
16 | riota2df.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
17 | 16 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝜒) |
18 | riota2df.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
19 | 18 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝐵) |
20 | 2, 5, 12, 15, 17, 19 | iota2df 6540 | . 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵)) |
21 | df-riota 7382 | . . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
22 | 21 | eqeq1i 2733 | . 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵) |
23 | 20, 22 | bitr4di 288 | 1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∃!weu 2557 Ⅎwnfc 2879 ∃!wreu 3372 ℩cio 6503 ℩crio 7381 |
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 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-reu 3375 df-v 3475 df-un 3954 df-in 3956 df-ss 3966 df-sn 4633 df-pr 4635 df-uni 4913 df-iota 6505 df-riota 7382 |
This theorem is referenced by: riota2f 7407 riotaeqimp 7409 riota5f 7411 mapdheq 41241 hdmap1eq 41314 hdmapval2lem 41344 |
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