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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 46406. In contrast to ovrcl 7445, 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 46382 | . . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩) | |
2 | 1 | eleq1i 2818 | . 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶) |
3 | afvvdm 46402 | . . 3 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 → ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) | |
4 | df-br 5142 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) | |
5 | aovprc.1 | . . . . 5 ⊢ Rel dom 𝐹 | |
6 | 5 | brrelex12i 5724 | . . . 4 ⊢ (𝐴dom 𝐹 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | 4, 6 | sylbir 234 | . . 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 216 | 1 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 class class class wbr 5141 dom cdm 5669 Rel wrel 5674 '''cafv 46378 ((caov 46379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6488 df-fun 6538 df-fv 6544 df-aiota 46346 df-dfat 46380 df-afv 46381 df-aov 46382 |
This theorem is referenced by: (None) |
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