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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 12004). By virtue of ndxslid 12023, 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 12010 | . . 3 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V) | |
4 | 2, 3 | mp1i 10 | . 2 ⊢ (𝐶 ∈ 𝑉 → 𝑆 ∈ V) |
5 | 2 | structfun 12016 | . . 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 4158 | . . . . 5 ⊢ (((𝐸‘ndx) ∈ ℕ ∧ 𝐶 ∈ 𝑉) → 〈(𝐸‘ndx), 𝐶〉 ∈ V) | |
10 | 8, 9 | mpan 421 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → 〈(𝐸‘ndx), 𝐶〉 ∈ V) |
11 | snssg 3664 | . . . 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 12040 | 1 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 {csn 3532 〈cop 3535 class class class wbr 3937 ◡ccnv 4546 Fun wfun 5125 ‘cfv 5131 ℕcn 8744 Struct cstr 11994 ndxcnx 11995 Slot cslot 11997 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fun 5133 df-fv 5139 df-struct 12000 df-slot 12002 |
This theorem is referenced by: strslfv3 12043 |
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