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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 15700. (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 15698 | . . . 4 ⊢ ndx = ( I ↾ ℕ) | |
2 | nnex 10903 | . . . . 5 ⊢ ℕ ∈ V | |
3 | resiexg 6994 | . . . . 5 ⊢ (ℕ ∈ V → ( I ↾ ℕ) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ↾ ℕ) ∈ V |
5 | 1, 4 | eqeltri 2684 | . . 3 ⊢ ndx ∈ V |
6 | ndxarg.1 | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
7 | 5, 6 | strfvn 15712 | . 2 ⊢ (𝐸‘ndx) = (ndx‘𝑁) |
8 | 1 | fveq1i 6104 | . 2 ⊢ (ndx‘𝑁) = (( I ↾ ℕ)‘𝑁) |
9 | ndxarg.2 | . . 3 ⊢ 𝑁 ∈ ℕ | |
10 | fvresi 6344 | . . 3 ⊢ (𝑁 ∈ ℕ → (( I ↾ ℕ)‘𝑁) = 𝑁) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (( I ↾ ℕ)‘𝑁) = 𝑁 |
12 | 7, 8, 11 | 3eqtri 2636 | 1 ⊢ (𝐸‘ndx) = 𝑁 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 I cid 4948 ↾ cres 5040 ‘cfv 5804 ℕcn 10897 ndxcnx 15692 Slot cslot 15694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-ndx 15698 df-slot 15699 |
This theorem is referenced by: ndxid 15716 basendx 15751 basendxnn 15752 resslem 15760 plusgndx 15803 2strstr 15809 2strstr1 15812 2strop1 15814 mulrndx 15821 starvndx 15827 scandx 15836 vscandx 15838 ipndx 15845 tsetndx 15863 plendx 15870 plendxOLD 15871 ocndx 15883 dsndx 15885 unifndx 15887 homndx 15897 ccondx 15899 slotsbhcdif 15903 oppglem 17603 mgplem 18317 opprlem 18451 sralem 18998 opsrbaslem 19298 opsrbaslemOLD 19299 zlmlem 19684 znbaslem 19705 znbaslemOLD 19706 tnglem 22254 itvndx 25139 lngndx 25140 ttglem 25556 cchhllem 25567 edgfndxnn 25669 baseltedgf 25671 resvlem 29162 hlhilslem 36248 basendxnmulrndx 41744 |
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