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Mirrors > Home > ILE Home > Th. List > ndxarg | GIF version |
Description: Get the numeric argument from a defined structure component extractor such as df-base 11965. (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 11962 | . . . 4 ⊢ ndx = ( I ↾ ℕ) | |
2 | nnex 8726 | . . . . 5 ⊢ ℕ ∈ V | |
3 | resiexg 4864 | . . . . 5 ⊢ (ℕ ∈ V → ( I ↾ ℕ) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ↾ ℕ) ∈ V |
5 | 1, 4 | eqeltri 2212 | . . 3 ⊢ ndx ∈ V |
6 | ndxarg.1 | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
7 | ndxarg.2 | . . 3 ⊢ 𝑁 ∈ ℕ | |
8 | 5, 6, 7 | strnfvn 11980 | . 2 ⊢ (𝐸‘ndx) = (ndx‘𝑁) |
9 | 1 | fveq1i 5422 | . 2 ⊢ (ndx‘𝑁) = (( I ↾ ℕ)‘𝑁) |
10 | fvresi 5613 | . . 3 ⊢ (𝑁 ∈ ℕ → (( I ↾ ℕ)‘𝑁) = 𝑁) | |
11 | 7, 10 | ax-mp 5 | . 2 ⊢ (( I ↾ ℕ)‘𝑁) = 𝑁 |
12 | 8, 9, 11 | 3eqtri 2164 | 1 ⊢ (𝐸‘ndx) = 𝑁 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 I cid 4210 ↾ cres 4541 ‘cfv 5123 ℕcn 8720 ndxcnx 11956 Slot cslot 11958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-inn 8721 df-ndx 11962 df-slot 11963 |
This theorem is referenced by: ndxid 11983 ndxslid 11984 strndxid 11987 basendx 12013 basendxnn 12014 plusgndx 12052 2strstrg 12059 2strbasg 12060 2stropg 12061 2strstr1g 12062 2strop1g 12064 basendxnplusgndx 12065 mulrndx 12069 basendxnmulrndx 12073 starvndx 12078 scandx 12086 vscandx 12089 ipndx 12097 tsetndx 12107 plendx 12114 dsndx 12117 |
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