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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-evalid | Structured version Visualization version GIF version |
Description: The evaluation at a set of the identity function is that set. (General form of ndxarg 16341.) The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.) |
Ref | Expression |
---|---|
bj-evalid | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resiexg 7482 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑉) ∈ V) | |
2 | bj-evalval 33985 | . . 3 ⊢ (( I ↾ 𝑉) ∈ V → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑉 ∈ 𝑊 → (Slot 𝐴‘( I ↾ 𝑉)) = (( I ↾ 𝑉)‘𝐴)) |
4 | fvresi 6805 | . 2 ⊢ (𝐴 ∈ 𝑉 → (( I ↾ 𝑉)‘𝐴) = 𝐴) | |
5 | 3, 4 | sylan9eq 2853 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 I cid 5354 ↾ cres 5452 ‘cfv 6232 Slot cslot 16315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-res 5462 df-iota 6196 df-fun 6234 df-fv 6240 df-slot 16320 |
This theorem is referenced by: bj-ndxarg 33987 bj-evalidval 33989 |
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