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| Mirrors > Home > MPE Home > Th. List > ovrcl | Structured version Visualization version GIF version | ||
| Description: Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.) |
| Ref | Expression |
|---|---|
| ovprc1.1 | ⊢ Rel dom 𝐹 |
| Ref | Expression |
|---|---|
| ovrcl | ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | n0i 4340 | . 2 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → ¬ (𝐴𝐹𝐵) = ∅) | |
| 2 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
| 3 | 2 | ovprc 7469 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
| 4 | 1, 3 | nsyl2 141 | 1 ⊢ (𝐶 ∈ (𝐴𝐹𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 dom cdm 5685 Rel wrel 5690 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: ghmquskerco 19302 smatrcl 33795 grimprop 47869 grimuhgr 47878 grimcnv 47879 grimco 47880 uhgrimisgrgric 47899 grlimprop 47951 grlimprop2 47953 |
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