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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 12521). By virtue of ndxslid 12540, 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 12527 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
4 | 2, 3 | mp1i 10 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V) |
5 | 2 | structfun 12533 | . . 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 4246 | . . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → 〈(𝐸‘ndx), 𝐶〉 ∈ V) | |
10 | 8, 9 | mpan 424 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ V) |
11 | snssg 3741 | . . . 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 12558 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 {csn 3607 〈cop 3610 class class class wbr 4018 ◡ccnv 4643 Fun wfun 5229 ‘cfv 5235 ℕcn 8950 Struct cstr 12511 ndxcnx 12512 Slot cslot 12514 |
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 615 ax-in2 616 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 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 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-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 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 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fv 5243 df-struct 12517 df-slot 12519 |
This theorem is referenced by: strslfv3 12561 cnfldbas 13885 cnfldadd 13886 cnfldmul 13887 cnfldcj 13888 |
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