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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 5306). Special case of riota2f 7140. (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 36094 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐴 ∈ V) → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)) |
6 | 1, 5 | mpan2 689 | . 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶)) |
7 | 6 | 3impib 1112 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝐷 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ℩crio 7115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 df-undef 7941 |
This theorem is referenced by: cdleme26e 37497 cdleme26eALTN 37499 cdleme26fALTN 37500 cdleme26f 37501 cdleme26f2ALTN 37502 cdleme26f2 37503 |
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