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| Mirrors > Home > MPE Home > Th. List > elfvov2 | Structured version Visualization version GIF version | ||
| Description: Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 4-Aug-2025.) |
| Ref | Expression |
|---|---|
| elfvov1.o | ⊢ Rel dom 𝑂 |
| elfvov1.s | ⊢ 𝑆 = (𝐼𝑂𝑅) |
| elfvov1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌)) |
| Ref | Expression |
|---|---|
| elfvov2 | ⊢ (𝜑 → 𝑅 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfvov1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌)) | |
| 2 | n0i 4340 | . . 3 ⊢ (𝑋 ∈ (𝑆‘𝑌) → ¬ (𝑆‘𝑌) = ∅) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ¬ (𝑆‘𝑌) = ∅) |
| 4 | elfvov1.s | . . . . 5 ⊢ 𝑆 = (𝐼𝑂𝑅) | |
| 5 | elfvov1.o | . . . . . 6 ⊢ Rel dom 𝑂 | |
| 6 | 5 | ovprc2 7471 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅) |
| 7 | 4, 6 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
| 8 | 7 | fveq1d 6908 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = (∅‘𝑌)) |
| 9 | 0fv 6950 | . . 3 ⊢ (∅‘𝑌) = ∅ | |
| 10 | 8, 9 | eqtrdi 2793 | . 2 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = ∅) |
| 11 | 3, 10 | nsyl2 141 | 1 ⊢ (𝜑 → 𝑅 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 dom cdm 5685 Rel wrel 5690 ‘cfv 6561 (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: ismhp 22144 mhprcl 22147 |
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