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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 5868 | . . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
2 | inss1 4202 | . . . . 5 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵 | |
3 | 1, 2 | eqsstri 3998 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵 |
4 | 3 | sseli 3960 | . . 3 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵) |
5 | 4 | con3i 157 | . 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵)) |
6 | ndmfv 6693 | . 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | |
7 | 5, 6 | syl 17 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ∅c0 4288 dom cdm 5548 ↾ cres 5550 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-res 5560 df-iota 6307 df-fv 6356 |
This theorem is referenced by: fveqres 6705 fvresval 32907 trpredlem1 32963 funpartfv 33303 |
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