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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 6032 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
2 | inss1 4245 | . . . 4 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵 | |
3 | 1, 2 | eqsstri 4030 | . . 3 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵 |
4 | 3 | sseli 3991 | . 2 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵) |
5 | ndmfv 6942 | . 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 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-res 5701 df-iota 6516 df-fv 6571 |
This theorem is referenced by: fveqres 6954 fvresval 7378 funpartfv 35927 setrec2lem1 48924 |
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