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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 5304). Special case of riota2f 7138. (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 36107 | . . 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 3138 Vcvv 3494 ℩crio 7113 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-undef 7939 |
This theorem is referenced by: cdleme26e 37510 cdleme26eALTN 37512 cdleme26fALTN 37513 cdleme26f 37514 cdleme26f2ALTN 37515 cdleme26f2 37516 |
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