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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ndxarg | Structured version Visualization version GIF version |
Description: Proof of ndxarg 16486 from bj-evalid 34378. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ndxarg.1 | ⊢ 𝐸 = Slot 𝑁 |
bj-ndxarg.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
bj-ndxarg | ⊢ (𝐸‘ndx) = 𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 11625 | . 2 ⊢ ℕ ∈ V | |
2 | bj-ndxarg.2 | . 2 ⊢ 𝑁 ∈ ℕ | |
3 | bj-ndxarg.1 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
4 | df-ndx 16464 | . . . 4 ⊢ ndx = ( I ↾ ℕ) | |
5 | 3, 4 | fveq12i 6657 | . . 3 ⊢ (𝐸‘ndx) = (Slot 𝑁‘( I ↾ ℕ)) |
6 | bj-evalid 34378 | . . 3 ⊢ ((ℕ ∈ V ∧ 𝑁 ∈ ℕ) → (Slot 𝑁‘( I ↾ ℕ)) = 𝑁) | |
7 | 5, 6 | syl5eq 2867 | . 2 ⊢ ((ℕ ∈ V ∧ 𝑁 ∈ ℕ) → (𝐸‘ndx) = 𝑁) |
8 | 1, 2, 7 | mp2an 690 | 1 ⊢ (𝐸‘ndx) = 𝑁 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3481 I cid 5440 ↾ cres 5538 ‘cfv 6336 ℕcn 11619 ndxcnx 16458 Slot cslot 16460 |
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 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-1cn 10576 ax-addcl 10578 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-ov 7140 df-om 7562 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-nn 11620 df-ndx 16464 df-slot 16465 |
This theorem is referenced by: (None) |
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