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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 5840 | . . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹) | |
2 | inss1 4155 | . . . 4 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵 | |
3 | 1, 2 | eqsstri 3949 | . . 3 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵 |
4 | 3 | sseli 3911 | . 2 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵) |
5 | ndmfv 6675 | . 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | |
6 | 4, 5 | nsyl5 162 | 1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ∅c0 4243 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-dm 5529 df-res 5531 df-iota 6283 df-fv 6332 |
This theorem is referenced by: fveqres 6687 fvresval 33123 trpredlem1 33179 funpartfv 33519 setrec2lem1 45223 |
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