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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 11971 | . . . 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 11973 | . . . . . . . . 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 995 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 ⊆ (𝑀...𝑁)) |
9 | 7 | simp1d 993 | . . . . . . . . 9 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) |
10 | 9 | simp1d 993 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
11 | fzssnn 9848 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ) | |
12 | 10, 11 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (𝑀...𝑁) ⊆ ℕ) |
13 | 8, 12 | sstrd 3107 | . . . . . 6 ⊢ (𝜑 → dom 𝑆 ⊆ ℕ) |
14 | 13, 4 | sseldd 3098 | . . . . 5 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
15 | 5, 3, 14 | strnfvnd 11979 | . . . 4 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
16 | strsetsid.f | . . . . 5 ⊢ (𝜑 → Fun 𝑆) | |
17 | funfvex 5438 | . . . . 5 ⊢ ((Fun 𝑆 ∧ (𝐸‘ndx) ∈ dom 𝑆) → (𝑆‘(𝐸‘ndx)) ∈ V) | |
18 | 16, 4, 17 | syl2anc 408 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) ∈ V) |
19 | 15, 18 | eqeltrd 2216 | . . 3 ⊢ (𝜑 → (𝐸‘𝑆) ∈ V) |
20 | setsvala 11990 | . . 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 1216 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉})) |
22 | 15 | opeq2d 3712 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), (𝐸‘𝑆)〉 = 〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉) |
23 | 22 | sneqd 3540 | . . 3 ⊢ (𝜑 → {〈(𝐸‘ndx), (𝐸‘𝑆)〉} = {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) |
24 | 23 | uneq2d 3230 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝐸‘𝑆)〉}) = ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉})) |
25 | nnssz 9071 | . . . . 5 ⊢ ℕ ⊆ ℤ | |
26 | 13, 25 | sstrdi 3109 | . . . 4 ⊢ (𝜑 → dom 𝑆 ⊆ ℤ) |
27 | zdceq 9126 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 = 𝑦) | |
28 | 27 | rgen2a 2486 | . . . 4 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 |
29 | ssralv 3161 | . . . . . 6 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) | |
30 | 29 | ralimdv 2500 | . . . . 5 ⊢ (dom 𝑆 ⊆ ℤ → (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℤ DECID 𝑥 = 𝑦 → ∀𝑥 ∈ ℤ ∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦)) |
31 | ssralv 3161 | . . . . 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 1422 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom 𝑆∀𝑦 ∈ dom 𝑆DECID 𝑥 = 𝑦) |
34 | funresdfunsndc 6402 | . . 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 1216 | . 2 ⊢ (𝜑 → ((𝑆 ↾ (V ∖ {(𝐸‘ndx)})) ∪ {〈(𝐸‘ndx), (𝑆‘(𝐸‘ndx))〉}) = 𝑆) |
36 | 21, 24, 35 | 3eqtrrd 2177 | 1 ⊢ (𝜑 → 𝑆 = (𝑆 sSet 〈(𝐸‘ndx), (𝐸‘𝑆)〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 819 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∀wral 2416 Vcvv 2686 ∖ cdif 3068 ∪ cun 3069 ⊆ wss 3071 ∅c0 3363 {csn 3527 〈cop 3530 class class class wbr 3929 dom cdm 4539 ↾ cres 4541 Fun wfun 5117 ‘cfv 5123 (class class class)co 5774 ≤ cle 7801 ℕcn 8720 ℤcz 9054 ...cfz 9790 Struct cstr 11955 ndxcnx 11956 sSet csts 11957 Slot cslot 11958 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-struct 11961 df-slot 11963 df-sets 11966 |
This theorem is referenced by: (None) |
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