Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aovrcl | Structured version Visualization version GIF version |
Description: Reverse closure for an operation value, analogous to afvvv 39874. In contrast to ovrcl 6584, elementhood of the operation's value in a set is required, not containing an element. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
aovprc.1 | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
aovrcl | ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aov 39847 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2679 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvdm 39870 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
4 | df-br 4584 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
5 | aovprc.1 | . . . . 5 ⊢ Rel dom 𝐹 | |
6 | brrelex12 5079 | . . . . 5 ⊢ ((Rel dom 𝐹 ∧ 𝐴dom 𝐹 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
7 | 5, 6 | mpan 702 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 4, 7 | sylbir 224 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | 3, 8 | syl 17 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
10 | 2, 9 | sylbi 206 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 '''cafv 39843 ((caov 39844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-fv 5812 df-dfat 39845 df-afv 39846 df-aov 39847 |
This theorem is referenced by: (None) |
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