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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 4345 | . . 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 7470 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅) |
7 | 4, 6 | eqtrid 2786 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅) |
8 | 7 | fveq1d 6908 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = (∅‘𝑌)) |
9 | 0fv 6950 | . . 3 ⊢ (∅‘𝑌) = ∅ | |
10 | 8, 9 | eqtrdi 2790 | . 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 1536 ∈ wcel 2105 Vcvv 3477 ∅c0 4338 dom cdm 5688 Rel wrel 5693 ‘cfv 6562 (class class class)co 7430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-dm 5698 df-iota 6515 df-fv 6570 df-ov 7433 |
This theorem is referenced by: ismhp 22161 mhprcl 22164 |
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