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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 5403). Special case of riota2f 7412. (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 38957 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)) |
| 6 | 1, 5 | mpan2 691 | . 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)) |
| 7 | 6 | 3impib 1117 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ℩crio 7387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-undef 8298 |
| This theorem is referenced by: cdleme26e 40361 cdleme26eALTN 40363 cdleme26fALTN 40364 cdleme26f 40365 cdleme26f2ALTN 40366 cdleme26f2 40367 |
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