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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 11649). By virtue of ndxslid 11668, 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 11655 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
4 | 2, 3 | mp1i 10 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V) |
5 | 2 | structfun 11661 | . . 3 ⊢ Fun ◡◡𝑆 |
6 | 5 | a1i 9 | . 2 ⊢ (𝐶 ∈ 𝑉 → Fun ◡◡𝑆) |
7 | strfv.n | . . 3 ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | |
8 | 1 | simpri 112 | . . . . 5 ⊢ (𝐸‘ndx) ∈ ℕ |
9 | opexg 4079 | . . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → 〈(𝐸‘ndx), 𝐶〉 ∈ V) | |
10 | 8, 9 | mpan 416 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ V) |
11 | snssg 3595 | . . . 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 167 | . 2 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
14 | id 19 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ 𝑉) | |
15 | 1, 4, 6, 13, 14 | strslfv2d 11685 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∈ wcel 1445 Vcvv 2633 ⊆ wss 3013 {csn 3466 〈cop 3469 class class class wbr 3867 ◡ccnv 4466 Fun wfun 5043 ‘cfv 5049 ℕcn 8520 Struct cstr 11639 ndxcnx 11640 Slot cslot 11642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-iota 5014 df-fun 5051 df-fv 5057 df-struct 11645 df-slot 11647 |
This theorem is referenced by: strslfv3 11688 |
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