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Mirrors > Home > ILE Home > Th. List > strslfv | GIF version |
Description: Extract a structure component 𝐶 (such as the base set) from a structure 𝑆 with a component extractor 𝐸 (such as the base set extractor df-base 12438). By virtue of ndxslid 12457, this can be done without having to refer to the hard-coded numeric index of 𝐸. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
strfv.s | ⊢ 𝑆 Struct 𝑋 |
strslfv.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
strfv.n | ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 |
Ref | Expression |
---|---|
strslfv | ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strslfv.e | . 2 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
2 | strfv.s | . . 3 ⊢ 𝑆 Struct 𝑋 | |
3 | structex 12444 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
4 | 2, 3 | mp1i 10 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V) |
5 | 2 | structfun 12450 | . . 3 ⊢ Fun ◡◡𝑆 |
6 | 5 | a1i 9 | . 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆) |
7 | strfv.n | . . 3 ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | |
8 | 1 | simpri 113 | . . . . 5 ⊢ (𝐸‘ndx) ∈ ℕ |
9 | opexg 4224 | . . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → 〈(𝐸‘ndx), 𝐶〉 ∈ V) | |
10 | 8, 9 | mpan 424 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ V) |
11 | snssg 3725 | . . . 4 ⊢ (〈(𝐸‘ndx), 𝐶〉 ∈ V → (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ↔ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆)) | |
12 | 10, 11 | syl 14 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 ↔ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆)) |
13 | 7, 12 | mpbiri 168 | . 2 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
14 | id 19 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉) | |
15 | 1, 4, 6, 13, 14 | strslfv2d 12474 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 {csn 3591 〈cop 3594 class class class wbr 4000 ◡ccnv 4621 Fun wfun 5205 ‘cfv 5211 ℕcn 8895 Struct cstr 12428 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-in1 614 ax-in2 615 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 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 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-struct 12434 df-slot 12436 |
This theorem is referenced by: strslfv3 12477 |
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