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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 12189 | . . . 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 12191 | . . . . . . . . 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 996 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 ⊆ (𝑀...𝑁)) |
9 | 7 | simp1d 994 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) |
10 | 9 | simp1d 994 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
11 | fzssnn 9965 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ) | |
12 | 10, 11 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀...𝑁) ⊆ ℕ) |
13 | 8, 12 | sstrd 3138 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 ⊆ ℕ) |
14 | 13, 4 | sseldd 3129 | . . . . 5 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
15 | 5, 3, 14 | strnfvnd 12197 | . . . 4 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
16 | strsetsid.f | . . . . 5 ⊢ (𝜑 → Fun 𝑆) | |
17 | funfvex 5484 | . . . . 5 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → (𝑆‘(𝐸‘ndx)) ∈ V) | |
18 | 16, 4, 17 | syl2anc 409 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) ∈ V) |
19 | 15, 18 | eqeltrd 2234 | . . 3 ⊢ (𝜑 → (𝐸‘𝑆) ∈ V) |
20 | setsvala 12208 | . . 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 1220 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
22 | 15 | opeq2d 3748 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
23 | 22 | sneqd 3573 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
24 | 23 | uneq2d 3261 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
25 | nnssz 9179 | . . . . 5 ⊢ ℕ ⊆ ℤ | |
26 | 13, 25 | sstrdi 3140 | . . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ ℤ) |
27 | zdceq 9234 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 = 𝑦) | |
28 | 27 | rgen2a 2511 | . . . 4 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 |
29 | ssralv 3192 | . . . . . 6 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
30 | 29 | ralimdv 2525 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
31 | ssralv 3192 | . . . . 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 1426 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦) |
34 | funresdfunsndc 6450 | . . 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 1220 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
36 | 21, 24, 35 | 3eqtrrd 2195 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 820 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 ∀wral 2435 Vcvv 2712 ∖ cdif 3099 ∪ cun 3100 ⊆ wss 3102 ∅c0 3394 {csn 3560 〈cop 3563 class class class wbr 3965 dom cdm 4585 ↾ cres 4587 Fun wfun 5163 ‘cfv 5169 (class class class)co 5821 ≤ cle 7908 ℕcn 8828 ℤcz 9162 ...cfz 9907 Struct cstr 12173 ndxcnx 12174 sSet csts 12175 Slot cslot 12176 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-addcom 7827 ax-addass 7829 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-0id 7835 ax-rnegex 7836 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-ltadd 7843 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-inn 8829 df-n0 9086 df-z 9163 df-uz 9435 df-fz 9908 df-struct 12179 df-slot 12181 df-sets 12184 |
This theorem is referenced by: (None) |
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