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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 12343. (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 12340 | . . . 4 ⊢ ndx = ( I ↾ ℕ) | |
2 | nnex 8854 | . . . . 5 ⊢ ℕ ∈ V | |
3 | resiexg 4923 | . . . . 5 ⊢ (ℕ ∈ V → ( I ↾ ℕ) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ↾ ℕ) ∈ V |
5 | 1, 4 | eqeltri 2237 | . . 3 ⊢ ndx ∈ V |
6 | ndxarg.1 | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
7 | ndxarg.2 | . . 3 ⊢ 𝑁 ∈ ℕ | |
8 | 5, 6, 7 | strnfvn 12358 | . 2 ⊢ (𝐸‘ndx) = (ndx‘𝑁) |
9 | 1 | fveq1i 5481 | . 2 ⊢ (ndx‘𝑁) = (( I ↾ ℕ)‘𝑁) |
10 | fvresi 5672 | . . 3 ⊢ (𝑁 ∈ ℕ → (( I ↾ ℕ)‘𝑁) = 𝑁) | |
11 | 7, 10 | ax-mp 5 | . 2 ⊢ (( I ↾ ℕ)‘𝑁) = 𝑁 |
12 | 8, 9, 11 | 3eqtri 2189 | 1 ⊢ (𝐸‘ndx) = 𝑁 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 Vcvv 2721 I cid 4260 ↾ cres 4600 ‘cfv 5182 ℕcn 8848 ndxcnx 12334 Slot cslot 12336 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fv 5190 df-inn 8849 df-ndx 12340 df-slot 12341 |
This theorem is referenced by: ndxid 12361 ndxslid 12362 strndxid 12365 basendx 12391 basendxnn 12392 plusgndx 12430 2strstrg 12437 2strbasg 12438 2stropg 12439 2strstr1g 12440 2strop1g 12442 basendxnplusgndx 12443 mulrndx 12447 basendxnmulrndx 12451 starvndx 12456 scandx 12464 vscandx 12467 ipndx 12475 tsetndx 12485 plendx 12492 dsndx 12495 |
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