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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-evalval | Structured version Visualization version GIF version |
Description: Value of the evaluation at a class. (Closed form of strfvnd 17153 and strfvn 17154). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.) |
Ref | Expression |
---|---|
bj-evalval | ⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3482 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | fveq1 6891 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴)) | |
3 | df-slot 17150 | . . 3 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | fvex 6905 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
5 | 2, 3, 4 | fvmpt 7000 | . 2 ⊢ (𝐹 ∈ V → (Slot 𝐴‘𝐹) = (𝐹‘𝐴)) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ‘cfv 6543 Slot cslot 17149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-slot 17150 |
This theorem is referenced by: bj-evalid 36612 bj-evalidval 36614 |
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