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Mirrors > Home > MPE Home > Th. List > riota2df | Structured version Visualization version GIF version |
Description: A deduction version of riota2f 7146. (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 484 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐵 ∈ 𝐴) |
3 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝜓) | |
4 | df-reu 3060 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | 3, 4 | sylib 221 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) |
6 | simpr 488 | . . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
7 | 2 | adantr 484 | . . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴) |
8 | 6, 7 | eqeltrd 2833 | . . . . 5 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) |
9 | 8 | biantrurd 536 | . . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))) |
10 | riota2df.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
11 | 10 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
12 | 9, 11 | bitr3d 284 | . . 3 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝜒)) |
13 | riota2df.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
14 | nfreu1 3272 | . . . 4 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜓 | |
15 | 13, 14 | nfan 1905 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) |
16 | riota2df.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
17 | 16 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝜒) |
18 | riota2df.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
19 | 18 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝐵) |
20 | 2, 5, 12, 15, 17, 19 | iota2df 6320 | . 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵)) |
21 | df-riota 7121 | . . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
22 | 21 | eqeq1i 2743 | . 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵) |
23 | 20, 22 | bitr4di 292 | 1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2113 ∃!weu 2569 Ⅎwnfc 2879 ∃!wreu 3055 ℩cio 6289 ℩crio 7120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-reu 3060 df-v 3399 df-sbc 3680 df-un 3846 df-in 3848 df-ss 3858 df-sn 4514 df-pr 4516 df-uni 4794 df-iota 6291 df-riota 7121 |
This theorem is referenced by: riota2f 7146 riotaeqimp 7148 riota5f 7150 mapdheq 39354 hdmap1eq 39427 hdmapval2lem 39457 |
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