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Mirrors > Home > MPE Home > Th. List > ndxarg | Structured version Visualization version GIF version |
Description: Get the numeric argument from a defined structure component extractor such as df-base 16481. (Contributed by Mario Carneiro, 6-Oct-2013.) |
Ref | Expression |
---|---|
ndxarg.1 | ⊢ 𝐸 = Slot 𝑁 |
ndxarg.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
ndxarg | ⊢ (𝐸‘ndx) = 𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ndx 16478 | . . . 4 ⊢ ndx = ( I ↾ ℕ) | |
2 | nnex 11636 | . . . . 5 ⊢ ℕ ∈ V | |
3 | resiexg 7611 | . . . . 5 ⊢ (ℕ ∈ V → ( I ↾ ℕ) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ↾ ℕ) ∈ V |
5 | 1, 4 | eqeltri 2907 | . . 3 ⊢ ndx ∈ V |
6 | ndxarg.1 | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
7 | 5, 6 | strfvn 16497 | . 2 ⊢ (𝐸‘ndx) = (ndx‘𝑁) |
8 | 1 | fveq1i 6664 | . 2 ⊢ (ndx‘𝑁) = (( I ↾ ℕ)‘𝑁) |
9 | ndxarg.2 | . . 3 ⊢ 𝑁 ∈ ℕ | |
10 | fvresi 6928 | . . 3 ⊢ (𝑁 ∈ ℕ → (( I ↾ ℕ)‘𝑁) = 𝑁) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (( I ↾ ℕ)‘𝑁) = 𝑁 |
12 | 7, 8, 11 | 3eqtri 2846 | 1 ⊢ (𝐸‘ndx) = 𝑁 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∈ wcel 2108 Vcvv 3493 I cid 5452 ↾ cres 5550 ‘cfv 6348 ℕcn 11630 ndxcnx 16472 Slot cslot 16474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-1cn 10587 ax-addcl 10589 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-nn 11631 df-ndx 16478 df-slot 16479 |
This theorem is referenced by: ndxid 16501 basendx 16539 basendxnn 16540 resslem 16549 plusgndx 16587 2strstr 16594 2strstr1 16597 2strop1 16599 basendxnplusgndx 16600 mulrndx 16607 basendxnmulrndx 16610 starvndx 16615 scandx 16624 vscandx 16626 ipndx 16633 tsetndx 16651 plendx 16658 ocndx 16665 dsndx 16667 unifndx 16669 homndx 16679 ccondx 16681 slotsbhcdif 16685 oppglem 18470 symgvalstruct 18517 mgplem 19236 opprlem 19370 rmodislmod 19694 sralem 19941 opsrbaslem 20250 zlmlem 20656 znbaslem 20677 tnglem 23241 itvndx 26218 lngndx 26219 ttglem 26654 cchhllem 26665 edgfndxnn 26769 baseltedgf 26771 resvlem 30897 hlhilslem 39066 |
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