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Mirrors > Home > ILE Home > Th. List > strnfvn | GIF version |
Description: Value of a structure
component extractor 𝐸. Normally, 𝐸 is a
defined constant symbol such as Base (df-base 12486) and 𝑁 is a
fixed integer such as 1. 𝑆 is a structure, i.e. a
specific
member of a class of structures.
Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12525. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
strnfvn.f | ⊢ 𝑆 ∈ V |
strnfvn.c | ⊢ 𝐸 = Slot 𝑁 |
strnfvn.n | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
strnfvn | ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strnfvn.c | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
2 | strnfvn.f | . . . 4 ⊢ 𝑆 ∈ V | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → 𝑆 ∈ V) |
4 | strnfvn.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → 𝑁 ∈ ℕ) |
6 | 1, 3, 5 | strnfvnd 12500 | . 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁)) |
7 | 6 | mptru 1373 | 1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ⊤wtru 1365 ∈ wcel 2160 Vcvv 2752 ‘cfv 5231 ℕcn 8937 Slot cslot 12479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-iota 5193 df-fun 5233 df-fv 5239 df-slot 12484 |
This theorem is referenced by: ndxarg 12503 strsl0 12529 baseval 12533 |
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