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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 11975) 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 12013. (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 11989 | . 2 ⊢ (⊤ → (𝐸‘𝑆) = (𝑆‘𝑁)) |
7 | 6 | mptru 1340 | 1 ⊢ (𝐸‘𝑆) = (𝑆‘𝑁) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 Vcvv 2686 ‘cfv 5123 ℕcn 8727 Slot cslot 11968 |
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 |
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-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-iota 5088 df-fun 5125 df-fv 5131 df-slot 11973 |
This theorem is referenced by: ndxarg 11992 strsl0 12017 baseval 12021 |
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