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Mirrors > Home > MPE Home > Th. List > Mathboxes > riotasv | Structured version Visualization version GIF version |
Description: Value of description binder 𝐷 for a single-valued class expression 𝐶(𝑦) (as in e.g. reusv2 5269). Special case of riota2f 7117. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) |
Ref | Expression |
---|---|
riotasv.1 | ⊢ 𝐴 ∈ V |
riotasv.2 | ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) |
Ref | Expression |
---|---|
riotasv | ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotasv.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | riotasv.2 | . . . . 5 ⊢ 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐷 ∈ 𝐴 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
4 | id 22 | . . . 4 ⊢ (𝐷 ∈ 𝐴 → 𝐷 ∈ 𝐴) | |
5 | 3, 4 | riotasvd 36252 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)) |
6 | 1, 5 | mpan2 690 | . 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)) |
7 | 6 | 3impib 1113 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ℩crio 7092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-undef 7922 |
This theorem is referenced by: cdleme26e 37655 cdleme26eALTN 37657 cdleme26fALTN 37658 cdleme26f 37659 cdleme26f2ALTN 37660 cdleme26f2 37661 |
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