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 43351. In contrast to ovrcl 7199, 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 43327 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) | |
2 | 1 | eleq1i 2905 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶) |
3 | afvvdm 43347 | . . 3 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
4 | df-br 5069 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) | |
5 | aovprc.1 | . . . . 5 ⊢ Rel dom 𝐹 | |
6 | 5 | brrelex12i 5609 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 4, 6 | sylbir 237 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ dom 𝐹 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 3, 7 | syl 17 | . 2 ⊢ ((𝐹'''〈𝐴, 𝐵〉) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
9 | 2, 8 | sylbi 219 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 〈cop 4575 class class class wbr 5068 dom cdm 5557 Rel wrel 5562 '''cafv 43323 ((caov 43324 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 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-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-aiota 43292 df-dfat 43325 df-afv 43326 df-aov 43327 |
This theorem is referenced by: (None) |
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