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Mirrors > Home > ILE Home > Th. List > strsetsid | GIF version |
Description: Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
strsetsid.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
strsetsid.s | ⊢ (𝜑 → 𝑆 Struct 〈𝑀, 𝑁〉) |
strsetsid.f | ⊢ (𝜑 → Fun 𝑆) |
strsetsid.d | ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) |
Ref | Expression |
---|---|
strsetsid | ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strsetsid.s | . . . 4 ⊢ (𝜑 → 𝑆 Struct 〈𝑀, 𝑁〉) | |
2 | structex 11569 | . . . 4 ⊢ (𝑆 Struct 〈𝑀, 𝑁〉 → 𝑆 ∈ V) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
4 | strsetsid.d | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ dom 𝑆) | |
5 | strsetsid.e | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
6 | isstructim 11571 | . . . . . . . . 9 ⊢ (𝑆 Struct 〈𝑀, 𝑁〉 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝑆 ∖ {∅}) ∧ dom 𝑆 ⊆ (𝑀...𝑁))) | |
7 | 1, 6 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝑆 ∖ {∅}) ∧ dom 𝑆 ⊆ (𝑀...𝑁))) |
8 | 7 | simp3d 958 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 ⊆ (𝑀...𝑁)) |
9 | 7 | simp1d 956 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) |
10 | 9 | simp1d 956 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
11 | fzssnn 9545 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ) | |
12 | 10, 11 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀...𝑁) ⊆ ℕ) |
13 | 8, 12 | sstrd 3038 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 ⊆ ℕ) |
14 | 13, 4 | sseldd 3029 | . . . . 5 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
15 | 5, 3, 14 | strnfvnd 11577 | . . . 4 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
16 | strsetsid.f | . . . . 5 ⊢ (𝜑 → Fun 𝑆) | |
17 | funfvex 5337 | . . . . 5 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → (𝑆‘(𝐸‘ndx)) ∈ V) | |
18 | 16, 4, 17 | syl2anc 404 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) ∈ V) |
19 | 15, 18 | eqeltrd 2165 | . . 3 ⊢ (𝜑 → (𝐸‘𝑆) ∈ V) |
20 | setsvala 11588 | . . 3 ⊢ ((𝑆 ∈ V ∧ (𝐸‘ndx) ∈ dom 𝑆 ∧ (𝐸‘𝑆) ∈ V) → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) | |
21 | 3, 4, 19, 20 | syl3anc 1175 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
22 | 15 | opeq2d 3637 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
23 | 22 | sneqd 3465 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
24 | 23 | uneq2d 3157 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
25 | nnssz 8830 | . . . . 5 ⊢ ℕ ⊆ ℤ | |
26 | 13, 25 | syl6ss 3040 | . . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ ℤ) |
27 | zdceq 8885 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 = 𝑦) | |
28 | 27 | rgen2a 2430 | . . . 4 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 |
29 | ssralv 3088 | . . . . . 6 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
30 | 29 | ralimdv 2443 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
31 | ssralv 3088 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
32 | 30, 31 | syld 45 | . . . 4 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
33 | 26, 28, 32 | mpisyl 1381 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦) |
34 | funresdfunsndc 6281 | . . 3 ⊢ ((∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦 ∧ Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) | |
35 | 33, 16, 4, 34 | syl3anc 1175 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
36 | 21, 24, 35 | 3eqtrrd 2126 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 781 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 ∀wral 2360 Vcvv 2622 ∖ cdif 2999 ∪ cun 3000 ⊆ wss 3002 ∅c0 3289 {csn 3452 〈cop 3455 class class class wbr 3853 dom cdm 4454 ↾ cres 4456 Fun wfun 5024 ‘cfv 5030 (class class class)co 5668 ≤ cle 7586 ℕcn 8485 ℤcz 8813 ...cfz 9487 Struct cstr 11553 ndxcnx 11554 sSet csts 11555 Slot cslot 11556 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-br 3854 df-opab 3908 df-mpt 3909 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-inn 8486 df-n0 8737 df-z 8814 df-uz 9083 df-fz 9488 df-struct 11559 df-slot 11561 df-sets 11564 |
This theorem is referenced by: (None) |
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