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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 12456 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | strndxid.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 1, 2 | ndxarg 12455 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
6 | 5, 2 | eqeltri 2250 | . . . 4 ⊢ (𝐸‘ndx) ∈ ℕ |
7 | 6 | a1i 9 | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
8 | 3, 4, 7 | strnfvnd 12452 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
9 | 8 | eqcomd 2183 | 1 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 ℕcn 8895 ndxcnx 12429 Slot cslot 12431 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fv 5219 df-inn 8896 df-ndx 12435 df-slot 12436 |
This theorem is referenced by: (None) |
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