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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvresval | Structured version Visualization version GIF version |
Description: The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.) |
Ref | Expression |
---|---|
fvresval | ⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmid 891 | . 2 ⊢ (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵) | |
2 | fvres 6683 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) | |
3 | nfvres 6700 | . . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | |
4 | 2, 3 | orim12i 905 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∅c0 4290 ↾ cres 5551 ‘cfv 6349 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-dm 5559 df-res 5561 df-iota 6308 df-fv 6357 |
This theorem is referenced by: sltres 33164 |
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