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Mirrors > Home > ILE Home > Th. List > strndxid | GIF version |
Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.) |
Ref | Expression |
---|---|
strndxid.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
strndxid.e | ⊢ 𝐸 = Slot 𝑁 |
strndxid.n | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
strndxid | ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strndxid.e | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | strndxid.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 11972 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | strndxid.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 1, 2 | ndxarg 11971 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
6 | 5, 2 | eqeltri 2210 | . . . 4 ⊢ (𝐸‘ndx) ∈ ℕ |
7 | 6 | a1i 9 | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
8 | 3, 4, 7 | strnfvnd 11968 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
9 | 8 | eqcomd 2143 | 1 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ‘cfv 5118 ℕcn 8713 ndxcnx 11945 Slot cslot 11947 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fv 5126 df-inn 8714 df-ndx 11951 df-slot 11952 |
This theorem is referenced by: (None) |
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