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| Mirrors > Home > MPE Home > Th. List > nfvres | Structured version Visualization version GIF version | ||
| Description: The value of a non-member of a restriction is the empty set. (An artifact of our function value definition.) (Contributed by NM, 13-Nov-1995.) |
| Ref | Expression |
|---|---|
| nfvres | ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmres 6030 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
| 2 | inss1 4237 | . . . 4 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵 | |
| 3 | 1, 2 | eqsstri 4030 | . . 3 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵 |
| 4 | 3 | sseli 3979 | . 2 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵) |
| 5 | ndmfv 6941 | . 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | |
| 6 | 4, 5 | nsyl5 159 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 |
| 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-in 3958 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-dm 5695 df-res 5697 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: fveqres 6953 fvresval 7378 funpartfv 35946 setrec2lem1 49212 |
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