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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 16504 and strfvn 16507). (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 3514 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | fveq1 6671 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓‘𝐴) = (𝐹‘𝐴)) | |
3 | df-slot 16489 | . . 3 ⊢ Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓‘𝐴)) | |
4 | fvex 6685 | . . 3 ⊢ (𝐹‘𝐴) ∈ V | |
5 | 2, 3, 4 | fvmpt 6770 | . 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 1537 ∈ wcel 2114 Vcvv 3496 ‘cfv 6357 Slot cslot 16484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-slot 16489 |
This theorem is referenced by: bj-evalid 34369 bj-evalidval 34371 |
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